diff --git a/Cargo.lock b/Cargo.lock
index debe8fbd6a..018e1a438c 100644
--- a/Cargo.lock
+++ b/Cargo.lock
@@ -41,15 +41,9 @@ checksum = "7825f6833612eb2414095684fcf6c635becf3ce97fe48cf6421321e93bfbd53c"
 
 [[package]]
 name = "arbitrary"
-version = "0.2.0"
+version = "0.4.0"
 source = "registry+https://github.com/rust-lang/crates.io-index"
-checksum = "64cf76cb6e2222ed0ea86b2b0ee2f71c96ec6edd5af42e84d59160e91b836ec4"
-
-[[package]]
-name = "arbitrary"
-version = "0.3.3"
-source = "registry+https://github.com/rust-lang/crates.io-index"
-checksum = "491d5e42b1a073ff1fc1e0a02744b3f8bee9cf4bfd552053cac36c64b879795d"
+checksum = "16971f2f0ce65c5cf2a1546cc6a0af102ecb11e265ddaa9433fb3e5bfdf676a4"
 dependencies = [
  "derive_arbitrary",
 ]
@@ -521,9 +515,9 @@ dependencies = [
 
 [[package]]
 name = "derive_arbitrary"
-version = "0.3.3"
+version = "0.4.0"
 source = "registry+https://github.com/rust-lang/crates.io-index"
-checksum = "dab2f0544254a47cabc58956cc7ebda74c3b796bb2761e3fe8f29fdde632ad95"
+checksum = "caedd6a71b6d00bdc458ec8ffbfd12689c1ee7ffa69ad9933310aaf2f08f18d8"
 dependencies = [
  "proc-macro2",
  "syn",
@@ -955,11 +949,11 @@ checksum = "eb147597cdf94ed43ab7a9038716637d2d1bf2bc571da995d0028dec06bd3018"
 
 [[package]]
 name = "libfuzzer-sys"
-version = "0.2.1"
+version = "0.3.1"
 source = "registry+https://github.com/rust-lang/crates.io-index"
-checksum = "3e969cd2be7a2aae0acbbe205da49134148db2287fb45a811bf441ed72f09a35"
+checksum = "fb789afcc589a08928d1e466087445ab740a0f70a2ee23d9349a0e3723d65e1b"
 dependencies = [
- "arbitrary 0.3.3",
+ "arbitrary",
  "cc",
 ]
 
@@ -2121,7 +2115,6 @@ dependencies = [
 name = "wasmtime-fuzz"
 version = "0.12.0"
 dependencies = [
- "arbitrary 0.2.0",
  "cranelift-codegen",
  "cranelift-reader",
  "cranelift-wasm",
@@ -2136,7 +2129,7 @@ name = "wasmtime-fuzzing"
 version = "0.12.0"
 dependencies = [
  "anyhow",
- "arbitrary 0.3.3",
+ "arbitrary",
  "binaryen",
  "env_logger 0.7.1",
  "log",
diff --git a/crates/fuzzing/Cargo.toml b/crates/fuzzing/Cargo.toml
index 74eec7c213..835b76edfc 100644
--- a/crates/fuzzing/Cargo.toml
+++ b/crates/fuzzing/Cargo.toml
@@ -8,7 +8,7 @@ version = "0.12.0"
 
 [dependencies]
 anyhow = "1.0.22"
-arbitrary = { version = "0.3.2", features = ["derive"] }
+arbitrary = { version = "0.4.0", features = ["derive"] }
 binaryen = "0.10.0"
 env_logger = "0.7.1"
 log = "0.4.8"
diff --git a/crates/fuzzing/src/generators.rs b/crates/fuzzing/src/generators.rs
index c3813a1b6e..f982b42844 100644
--- a/crates/fuzzing/src/generators.rs
+++ b/crates/fuzzing/src/generators.rs
@@ -44,6 +44,10 @@ impl Arbitrary for WasmOptTtf {
         let wasm = module.write();
         Ok(WasmOptTtf { wasm })
     }
+
+    fn size_hint(depth: usize) -> (usize, Option<usize>) {
+        <Vec<u8> as Arbitrary>::size_hint(depth)
+    }
 }
 
 /// A description of configuration options that we should do differential
diff --git a/crates/fuzzing/src/generators/api.rs b/crates/fuzzing/src/generators/api.rs
index 9ffe8c5e26..3def5aa9a2 100644
--- a/crates/fuzzing/src/generators/api.rs
+++ b/crates/fuzzing/src/generators/api.rs
@@ -48,12 +48,20 @@ use ApiCall::*;
 #[derive(Default)]
 struct Scope {
     id_counter: usize,
-    predicted_rss: usize,
+
     /// Map from a module id to the predicted amount of rss it will take to
     /// instantiate.
     modules: BTreeMap<usize, usize>,
+
     /// Map from an instance id to the amount of rss it's expected to be using.
     instances: BTreeMap<usize, usize>,
+
+    /// The rough predicted maximum RSS of executing all of our generated API
+    /// calls thus far.
+    predicted_rss: usize,
+
+    /// The number of calls of an exported function from an instance.
+    num_export_calls: usize,
 }
 
 impl Scope {
@@ -83,6 +91,11 @@ impl Arbitrary for ApiCalls {
         let mut scope = Scope::default();
         let max_rss = 1 << 30; // 1GB
 
+        // Calling an exported function of a `wasm-opt -ttf` module tends to
+        // take about 20ms. Limit their number to 100, or ~2s, so that we don't
+        // get too close to our 3s timeout.
+        let max_export_calls = 100;
+
         for _ in 0..input.arbitrary_len::<ApiCall>()? {
             let mut choices: Vec<fn(_, &mut Scope) -> arbitrary::Result<ApiCall>> = vec![];
 
@@ -122,8 +135,12 @@ impl Arbitrary for ApiCalls {
                     Ok(InstanceDrop { id })
                 });
             }
-            if swarm.call_exported_func && !scope.instances.is_empty() {
+            if swarm.call_exported_func
+                && scope.num_export_calls < max_export_calls
+                && !scope.instances.is_empty()
+            {
                 choices.push(|input, scope| {
+                    scope.num_export_calls += 1;
                     let instances: Vec<_> = scope.instances.keys().collect();
                     let instance = **input.choose(&instances)?;
                     let nth = usize::arbitrary(input)?;
@@ -140,6 +157,25 @@ impl Arbitrary for ApiCalls {
 
         Ok(ApiCalls { calls })
     }
+
+    fn size_hint(depth: usize) -> (usize, Option<usize>) {
+        arbitrary::size_hint::recursion_guard(depth, |depth| {
+            arbitrary::size_hint::or(
+                // This is the stuff we unconditionally need, which affects the
+                // minimum size.
+                arbitrary::size_hint::and(
+                    <Swarm as Arbitrary>::size_hint(depth),
+                    // `arbitrary_config` uses two bools when
+                    // `swarm.config_debug_info` is true.
+                    <(bool, bool) as Arbitrary>::size_hint(depth),
+                ),
+                // We can generate arbitrary `WasmOptTtf` instances, which have
+                // no upper bound on the number of bytes they consume. This sets
+                // the upper bound to `None`.
+                <super::WasmOptTtf as Arbitrary>::size_hint(depth),
+            )
+        })
+    }
 }
 
 fn arbitrary_config(
diff --git a/fuzz/Cargo.toml b/fuzz/Cargo.toml
index 6a24baa0d8..2c9eb4dc56 100644
--- a/fuzz/Cargo.toml
+++ b/fuzz/Cargo.toml
@@ -9,11 +9,10 @@ publish = false
 cargo-fuzz = true
 
 [dependencies]
-arbitrary = "0.2.0"
 cranelift-codegen = { path = "../cranelift/codegen" }
 cranelift-reader = { path = "../cranelift/reader" }
 cranelift-wasm = { path = "../cranelift/wasm" }
-libfuzzer-sys = "0.2.1"
+libfuzzer-sys = "0.3.1"
 target-lexicon = "0.10"
 wasmtime = { path = "../crates/api" }
 wasmtime-fuzzing = { path = "../crates/fuzzing" }