diff --git a/containers-tests/benchmarks/Map.hs b/containers-tests/benchmarks/Map.hs
index b53a4914d..0e324e556 100644
--- a/containers-tests/benchmarks/Map.hs
+++ b/containers-tests/benchmarks/Map.hs
@@ -95,6 +95,8 @@ main = do
         , bench "fromDistinctDescList" $ whnf M.fromDistinctDescList elems_rev
         , bench "fromDistinctDescList:fusion" $ whnf (\n -> M.fromDistinctDescList [(i,i) | i <- [n,n-1..1]]) bound
         , bench "minView" $ whnf (\m' -> case M.minViewWithKey m' of {Nothing -> 0; Just ((k,v),m'') -> k+v+M.size m''}) (M.fromAscList $ zip [1..10::Int] [100..110::Int])
+        , bench "eq" $ whnf (\m' -> m' == m') m -- worst case, compares everything
+        , bench "compare" $ whnf (\m' -> compare m' m') m -- worst case, compares everything
         ]
   where
     bound = 2^12
diff --git a/containers-tests/benchmarks/Set.hs b/containers-tests/benchmarks/Set.hs
index f65e2a620..265117e9c 100644
--- a/containers-tests/benchmarks/Set.hs
+++ b/containers-tests/benchmarks/Set.hs
@@ -55,6 +55,8 @@ main = do
         , bench "member.powerSet (16)" $ whnf (\ s -> all (flip S.member s) s) (S.powerSet (S.fromList [1..16]))
         , bench "member.powerSet (17)" $ whnf (\ s -> all (flip S.member s) s) (S.powerSet (S.fromList [1..17]))
         , bench "member.powerSet (18)" $ whnf (\ s -> all (flip S.member s) s) (S.powerSet (S.fromList [1..18]))
+        , bench "eq" $ whnf (\s' -> s' == s') s -- worst case, compares everything
+        , bench "compare" $ whnf (\s' -> compare s' s') s -- worst case, compares everything
         ]
   where
     bound = 2^12
diff --git a/containers-tests/containers-tests.cabal b/containers-tests/containers-tests.cabal
index d6857040d..9a098820d 100644
--- a/containers-tests/containers-tests.cabal
+++ b/containers-tests/containers-tests.cabal
@@ -124,6 +124,7 @@ library
     Utils.Containers.Internal.PtrEquality
     Utils.Containers.Internal.State
     Utils.Containers.Internal.StrictMaybe
+    Utils.Containers.Internal.EqOrdUtil
 
   if impl(ghc)
     other-modules:
diff --git a/containers-tests/tests/map-properties.hs b/containers-tests/tests/map-properties.hs
index b2f01f1b7..b6d873c44 100644
--- a/containers-tests/tests/map-properties.hs
+++ b/containers-tests/tests/map-properties.hs
@@ -33,7 +33,7 @@ import Test.Tasty
 import Test.Tasty.HUnit
 import Test.Tasty.QuickCheck
 import Test.QuickCheck.Function (apply)
-import Test.QuickCheck.Poly (A, B)
+import Test.QuickCheck.Poly (A, B, OrdA)
 import Control.Arrow (first)
 
 default (Int)
@@ -250,6 +250,8 @@ main = defaultMain $ testGroup "map-properties"
          , testProperty "splitAt"              prop_splitAt
          , testProperty "lookupMin"            prop_lookupMin
          , testProperty "lookupMax"            prop_lookupMax
+         , testProperty "eq"                   prop_eq
+         , testProperty "compare"              prop_compare
          ]
 
 {--------------------------------------------------------------------
@@ -1636,3 +1638,9 @@ prop_fromArgSet :: [(Int, Int)] -> Bool
 prop_fromArgSet ys =
   let xs = List.nubBy ((==) `on` fst) ys
   in fromArgSet (Set.fromList $ List.map (uncurry Arg) xs) == fromList xs
+
+prop_eq :: Map Int A -> Map Int A -> Property
+prop_eq m1 m2 = (m1 == m2) === (toList m1 == toList m2)
+
+prop_compare :: Map Int OrdA -> Map Int OrdA -> Property
+prop_compare m1 m2 = compare m1 m2 === compare (toList m1) (toList m2)
diff --git a/containers-tests/tests/set-properties.hs b/containers-tests/tests/set-properties.hs
index aeee9e584..82d30f0c6 100644
--- a/containers-tests/tests/set-properties.hs
+++ b/containers-tests/tests/set-properties.hs
@@ -110,6 +110,8 @@ main = defaultMain $ testGroup "set-properties"
                    , testProperty "strict foldr"         prop_strictFoldr'
                    , testProperty "strict foldl"         prop_strictFoldl'
 #endif
+                   , testProperty "eq" prop_eq
+                   , testProperty "compare" prop_compare
                    ]
 
 -- A type with a peculiar Eq instance designed to make sure keys
@@ -730,3 +732,9 @@ prop_strictFoldr' m = whnfHasNoThunks (foldr' (:) [] m)
 prop_strictFoldl' :: Set Int -> Property
 prop_strictFoldl' m = whnfHasNoThunks (foldl' (flip (:)) [] m)
 #endif
+
+prop_eq :: Set Int -> Set Int -> Property
+prop_eq s1 s2 = (s1 == s2) === (toList s1 == toList s2)
+
+prop_compare :: Set Int -> Set Int -> Property
+prop_compare s1 s2 = compare s1 s2 === compare (toList s1) (toList s2)
diff --git a/containers/containers.cabal b/containers/containers.cabal
index 1a89484b1..3ab69d3ac 100644
--- a/containers/containers.cabal
+++ b/containers/containers.cabal
@@ -80,6 +80,7 @@ Library
         Utils.Containers.Internal.StrictMaybe
         Utils.Containers.Internal.PtrEquality
         Utils.Containers.Internal.Coercions
+        Utils.Containers.Internal.EqOrdUtil
     if impl(ghc)
       other-modules:
         Utils.Containers.Internal.TypeError
diff --git a/containers/src/Data/Map/Internal.hs b/containers/src/Data/Map/Internal.hs
index d60af97b0..b0f29021a 100644
--- a/containers/src/Data/Map/Internal.hs
+++ b/containers/src/Data/Map/Internal.hs
@@ -401,6 +401,7 @@ import Utils.Containers.Internal.PtrEquality (ptrEq)
 import Utils.Containers.Internal.StrictPair
 import Utils.Containers.Internal.StrictMaybe
 import Utils.Containers.Internal.BitQueue
+import Utils.Containers.Internal.EqOrdUtil (EqM(..), OrdM(..))
 #ifdef DEFINE_ALTERF_FALLBACK
 import Utils.Containers.Internal.BitUtil (wordSize)
 #endif
@@ -4118,6 +4119,31 @@ deleteFindMax t = case maxViewWithKey t of
   Nothing -> (error "Map.deleteFindMax: can not return the maximal element of an empty map", Tip)
   Just res -> res
 
+{--------------------------------------------------------------------
+  Iterator
+--------------------------------------------------------------------}
+
+-- See Note [Iterator] in Data.Set.Internal
+
+iterDown :: Map k a -> Stack k a -> Stack k a
+iterDown (Bin _ kx x l r) stk = iterDown l (Push kx x r stk)
+iterDown Tip stk = stk
+
+-- Create an iterator from a Map, starting at the smallest key.
+iterator :: Map k a -> Stack k a
+iterator m = iterDown m Nada
+
+-- Get the next key-value and the remaining iterator.
+iterNext :: Stack k a -> Maybe (StrictPair (KeyValue k a) (Stack k a))
+iterNext (Push kx x r stk) = Just $! KeyValue kx x :*: iterDown r stk
+iterNext Nada = Nothing
+{-# INLINE iterNext #-}
+
+-- Whether there are no more key-values in the iterator.
+iterNull :: Stack k a -> Bool
+iterNull (Push _ _ _ _) = False
+iterNull Nada = True
+
 {--------------------------------------------------------------------
   [balance l x r] balances two trees with value x.
   The sizes of the trees should balance after decreasing the
@@ -4284,41 +4310,69 @@ bin k x l r
 
 
 {--------------------------------------------------------------------
-  Eq converts the tree to a list. In a lazy setting, this
-  actually seems one of the faster methods to compare two trees
-  and it is certainly the simplest :-)
+  Eq
 --------------------------------------------------------------------}
+
 instance (Eq k,Eq a) => Eq (Map k a) where
-  t1 == t2  = (size t1 == size t2) && (toAscList t1 == toAscList t2)
+  m1 == m2 = liftEq2 (==) (==) m1 m2
+  {-# INLINABLE (==) #-}
 
-{--------------------------------------------------------------------
-  Ord
---------------------------------------------------------------------}
+-- | @since 0.5.9
+instance Eq k => Eq1 (Map k) where
+  liftEq = liftEq2 (==)
+  {-# INLINE liftEq #-}
 
-instance (Ord k, Ord v) => Ord (Map k v) where
-    compare m1 m2 = compare (toAscList m1) (toAscList m2)
+-- | @since 0.5.9
+instance Eq2 Map where
+  liftEq2 keq eq m1 m2 = size m1 == size m2 && sameSizeLiftEq2 keq eq m1 m2
+  {-# INLINE liftEq2 #-}
+
+-- Assumes the maps are of equal size to skip the final check
+sameSizeLiftEq2
+  :: (ka -> kb -> Bool) -> (a -> b -> Bool) -> Map ka a -> Map kb b -> Bool
+sameSizeLiftEq2 keq eq m1 m2 =
+  case runEqM (foldMapWithKey f m1) (iterator m2) of e :*: _ -> e
+  where
+    f kx x = EqM $ \it -> case iterNext it of
+      Nothing -> False :*: it
+      Just (KeyValue ky y :*: it') -> (keq kx ky && eq x y) :*: it'
+{-# INLINE sameSizeLiftEq2 #-}
 
 {--------------------------------------------------------------------
-  Lifted instances
+  Ord
 --------------------------------------------------------------------}
 
--- | @since 0.5.9
-instance Eq2 Map where
-    liftEq2 eqk eqv m n =
-        size m == size n && liftEq (liftEq2 eqk eqv) (toList m) (toList n)
+instance (Ord k, Ord v) => Ord (Map k v) where
+  compare m1 m2 = liftCmp2 compare compare m1 m2
+  {-# INLINABLE compare #-}
 
 -- | @since 0.5.9
-instance Eq k => Eq1 (Map k) where
-    liftEq = liftEq2 (==)
+instance Ord k => Ord1 (Map k) where
+  liftCompare = liftCmp2 compare
+  {-# INLINE liftCompare #-}
 
 -- | @since 0.5.9
 instance Ord2 Map where
-    liftCompare2 cmpk cmpv m n =
-        liftCompare (liftCompare2 cmpk cmpv) (toList m) (toList n)
+  liftCompare2 = liftCmp2
+  {-# INLINE liftCompare2 #-}
+
+liftCmp2
+  :: (ka -> kb -> Ordering)
+  -> (a -> b -> Ordering)
+  -> Map ka a
+  -> Map kb b
+  -> Ordering
+liftCmp2 kcmp cmp m1 m2 = case runOrdM (foldMapWithKey f m1) (iterator m2) of
+  o :*: it -> o <> if iterNull it then EQ else LT
+  where
+    f kx x = OrdM $ \it -> case iterNext it of
+      Nothing -> GT :*: it
+      Just (KeyValue ky y :*: it') -> (kcmp kx ky <> cmp x y) :*: it'
+{-# INLINE liftCmp2 #-}
 
--- | @since 0.5.9
-instance Ord k => Ord1 (Map k) where
-    liftCompare = liftCompare2 compare
+{--------------------------------------------------------------------
+  Lifted instances
+--------------------------------------------------------------------}
 
 -- | @since 0.5.9
 instance Show2 Map where
diff --git a/containers/src/Data/Set/Internal.hs b/containers/src/Data/Set/Internal.hs
index f1ec29c3a..0d79ac9ce 100644
--- a/containers/src/Data/Set/Internal.hs
+++ b/containers/src/Data/Set/Internal.hs
@@ -252,6 +252,7 @@ import Control.DeepSeq (NFData(rnf))
 
 import Utils.Containers.Internal.StrictPair
 import Utils.Containers.Internal.PtrEquality
+import Utils.Containers.Internal.EqOrdUtil (EqM(..), OrdM(..))
 
 #if __GLASGOW_HASKELL__
 import GHC.Exts ( build, lazy )
@@ -1272,19 +1273,90 @@ foldl'Stack f = go
 {-# INLINE foldl'Stack #-}
 
 {--------------------------------------------------------------------
-  Eq converts the set to a list. In a lazy setting, this
-  actually seems one of the faster methods to compare two trees
-  and it is certainly the simplest :-)
+  Iterator
 --------------------------------------------------------------------}
+
+-- Note [Iterator]
+-- ~~~~~~~~~~~~~~~
+-- Iteration, using a Stack as an iterator, is an efficient way to consume a Set
+-- one element at a time. Alternately, this may be done by toAscList. toAscList
+-- when consumed via List.foldr will rewrite to Set.foldr (thanks to rewrite
+-- rules), which is quite efficient. However, sometimes that is not possible,
+-- such as in the second arg of '==' or 'compare', where manifesting the list
+-- cons cells is unavoidable and makes things slower.
+--
+-- Concretely, compare on Set Int using toAscList takes ~21% more time compared
+-- to using Iterator, on GHC 9.6.3.
+--
+-- The heart of this implementation is the `iterDown` function. It walks down
+-- the left spine of the tree, pushing the value and right child on the stack,
+-- until a Tip is reached. The next value is now at the top of the stack. To get
+-- to the value after that, `iterDown` is called again with the right child and
+-- the remaining stack.
+
+iterDown :: Set a -> Stack a -> Stack a
+iterDown (Bin _ x l r) stk = iterDown l (Push x r stk)
+iterDown Tip stk = stk
+
+-- Create an iterator from a Set, starting at the smallest element.
+iterator :: Set a -> Stack a
+iterator s = iterDown s Nada
+
+-- Get the next element and the remaining iterator.
+iterNext :: Stack a -> Maybe (StrictPair a (Stack a))
+iterNext (Push x r stk) = Just $! x :*: iterDown r stk
+iterNext Nada = Nothing
+{-# INLINE iterNext #-}
+
+-- Whether there are no more elements in the iterator.
+iterNull :: Stack a -> Bool
+iterNull (Push _ _ _) = False
+iterNull Nada = True
+
+{--------------------------------------------------------------------
+  Eq
+--------------------------------------------------------------------}
+
 instance Eq a => Eq (Set a) where
-  t1 == t2  = (size t1 == size t2) && (toAscList t1 == toAscList t2)
+  s1 == s2 = liftEq (==) s1 s2
+  {-# INLINABLE (==) #-}
+
+-- | @since 0.5.9
+instance Eq1 Set where
+  liftEq eq s1 s2 = size s1 == size s2 && sameSizeLiftEq eq s1 s2
+  {-# INLINE liftEq #-}
+
+-- Assumes the sets are of equal size to skip the final check.
+sameSizeLiftEq :: (a -> b -> Bool) -> Set a -> Set b -> Bool
+sameSizeLiftEq eq s1 s2 =
+  case runEqM (foldMap f s1) (iterator s2) of e :*: _ -> e
+  where
+    f x = EqM $ \it -> case iterNext it of
+      Nothing -> False :*: it
+      Just (y :*: it') -> eq x y :*: it'
+{-# INLINE sameSizeLiftEq #-}
 
 {--------------------------------------------------------------------
   Ord
 --------------------------------------------------------------------}
 
 instance Ord a => Ord (Set a) where
-    compare s1 s2 = compare (toAscList s1) (toAscList s2)
+  compare s1 s2 = liftCmp compare s1 s2
+  {-# INLINABLE compare #-}
+
+-- | @since 0.5.9
+instance Ord1 Set where
+  liftCompare = liftCmp
+  {-# INLINE liftCompare #-}
+
+liftCmp :: (a -> b -> Ordering) -> Set a -> Set b -> Ordering
+liftCmp cmp s1 s2 = case runOrdM (foldMap f s1) (iterator s2) of
+  o :*: it -> o <> if iterNull it then EQ else LT
+  where
+    f x = OrdM $ \it -> case iterNext it of
+      Nothing -> GT :*: it
+      Just (y :*: it') -> cmp x y :*: it'
+{-# INLINE liftCmp #-}
 
 {--------------------------------------------------------------------
   Show
@@ -1293,16 +1365,6 @@ instance Show a => Show (Set a) where
   showsPrec p xs = showParen (p > 10) $
     showString "fromList " . shows (toList xs)
 
--- | @since 0.5.9
-instance Eq1 Set where
-    liftEq eq m n =
-        size m == size n && liftEq eq (toList m) (toList n)
-
--- | @since 0.5.9
-instance Ord1 Set where
-    liftCompare cmp m n =
-        liftCompare cmp (toList m) (toList n)
-
 -- | @since 0.5.9
 instance Show1 Set where
     liftShowsPrec sp sl d m =
diff --git a/containers/src/Utils/Containers/Internal/EqOrdUtil.hs b/containers/src/Utils/Containers/Internal/EqOrdUtil.hs
new file mode 100644
index 000000000..58d39d8bd
--- /dev/null
+++ b/containers/src/Utils/Containers/Internal/EqOrdUtil.hs
@@ -0,0 +1,38 @@
+{-# LANGUAGE CPP #-}
+module Utils.Containers.Internal.EqOrdUtil
+  ( EqM(..)
+  , OrdM(..)
+  ) where
+
+#if !MIN_VERSION_base(4,11,0)
+import Data.Semigroup (Semigroup(..))
+#endif
+import Utils.Containers.Internal.StrictPair
+
+newtype EqM a = EqM { runEqM :: a -> StrictPair Bool a }
+
+-- | Composes left-to-right, short-circuits on False
+instance Semigroup (EqM a) where
+  f <> g = EqM $ \x -> case runEqM f x of
+    r@(e :*: x') -> if e then runEqM g x' else r
+
+instance Monoid (EqM a) where
+  mempty = EqM (True :*:)
+#if !MIN_VERSION_base(4,11,0)
+  mappend = (<>)
+#endif
+
+newtype OrdM a = OrdM { runOrdM :: a -> StrictPair Ordering a }
+
+-- | Composes left-to-right, short-circuits on non-EQ
+instance Semigroup (OrdM a) where
+  f <> g = OrdM $ \x -> case runOrdM f x of
+    r@(o :*: x') -> case o of
+      EQ -> runOrdM g x'
+      _ -> r
+
+instance Monoid (OrdM a) where
+  mempty = OrdM (EQ :*:)
+#if !MIN_VERSION_base(4,11,0)
+  mappend = (<>)
+#endif