diff --git a/uni-stark/src/check_constraints.rs b/uni-stark/src/check_constraints.rs
index 9014dbc3e..6a7b48cd1 100644
--- a/uni-stark/src/check_constraints.rs
+++ b/uni-stark/src/check_constraints.rs
@@ -151,11 +151,11 @@ impl<'a, F: Field> MultistageAirBuilder for DebugConstraintBuilder<'a, F> {
         self.public_values[stage]
     }
 
-    fn multi_stage(&self, stage: usize) -> Self::M {
+    fn stage_trace(&self, stage: usize) -> Self::M {
         self.stages[stage]
     }
 
-    fn challenges(&self, stage: usize) -> &[Self::Expr] {
+    fn stage_challenges(&self, stage: usize) -> &[Self::Expr] {
         self.challenges[stage]
     }
 }
diff --git a/uni-stark/src/folder.rs b/uni-stark/src/folder.rs
index 1d353251b..307337922 100644
--- a/uni-stark/src/folder.rs
+++ b/uni-stark/src/folder.rs
@@ -80,11 +80,11 @@ impl<'a, SC: StarkGenericConfig> AirBuilderWithPublicValues for ProverConstraint
 impl<'a, SC: StarkGenericConfig> MultistageAirBuilder for ProverConstraintFolder<'a, SC> {
     type Challenge = Val<SC>;
 
-    fn multi_stage(&self, stage: usize) -> <Self as AirBuilder>::M {
+    fn stage_trace(&self, stage: usize) -> <Self as AirBuilder>::M {
         self.stages[stage].clone()
     }
 
-    fn challenges(&self, stage: usize) -> &[Self::Challenge] {
+    fn stage_challenges(&self, stage: usize) -> &[Self::Challenge] {
         &self.challenges[stage]
     }
     fn stage_public_values(&self, stage: usize) -> &[Self::PublicVar] {
@@ -142,11 +142,11 @@ impl<'a, SC: StarkGenericConfig> AirBuilderWithPublicValues for VerifierConstrai
 impl<'a, SC: StarkGenericConfig> MultistageAirBuilder for VerifierConstraintFolder<'a, SC> {
     type Challenge = Val<SC>;
 
-    fn multi_stage(&self, stage: usize) -> <Self as AirBuilder>::M {
+    fn stage_trace(&self, stage: usize) -> <Self as AirBuilder>::M {
         self.stages[stage]
     }
 
-    fn challenges(&self, stage: usize) -> &[Self::Challenge] {
+    fn stage_challenges(&self, stage: usize) -> &[Self::Challenge] {
         &self.challenges[stage]
     }
     fn stage_public_values(&self, stage: usize) -> &[Self::PublicVar] {
diff --git a/uni-stark/src/prover.rs b/uni-stark/src/prover.rs
index b4e0bc64e..255a32fa5 100644
--- a/uni-stark/src/prover.rs
+++ b/uni-stark/src/prover.rs
@@ -17,7 +17,8 @@ use tracing::{info_span, instrument};
 use crate::symbolic_builder::{get_log_quotient_degree, SymbolicAirBuilder};
 use crate::traits::MultiStageAir;
 use crate::{
-    Commitments, Domain, OpenedValues, PackedChallenge, PackedVal, ProcessedStage, Proof, ProverConstraintFolder, StarkGenericConfig, StarkProvingKey, Val
+    Commitments, Domain, OpenedValues, PackedChallenge, PackedVal, ProcessedStage, Proof,
+    ProverConstraintFolder, StarkGenericConfig, StarkProvingKey, Val,
 };
 
 struct UnusedCallback;
@@ -102,7 +103,7 @@ where
     let mut state: ProverState<SC> = ProverState::new(pcs, trace_domain, challenger);
     let mut stage = Stage {
         trace: stage_0_trace,
-        challenge_count: <A as MultiStageAir<SymbolicAirBuilder<_>>>::challenge_count(air, 0),
+        challenge_count: <A as MultiStageAir<SymbolicAirBuilder<_>>>::stage_challenge_count(air, 0),
         public_values: stage_0_public_values.to_owned(),
     };
 
@@ -122,7 +123,7 @@ where
         // go to the next stage
         stage = Stage {
             trace,
-            challenge_count: <A as MultiStageAir<SymbolicAirBuilder<_>>>::challenge_count(
+            challenge_count: <A as MultiStageAir<SymbolicAirBuilder<_>>>::stage_challenge_count(
                 air,
                 stage_id as u32,
             ),
@@ -407,7 +408,6 @@ where
         .collect()
 }
 
-/// Updating with each new trace in every stage
 pub struct ProverState<'a, SC: StarkGenericConfig> {
     pub(crate) processed_stages: Vec<ProcessedStage<SC>>,
     pub(crate) challenger: &'a mut SC::Challenger,
diff --git a/uni-stark/src/symbolic_builder.rs b/uni-stark/src/symbolic_builder.rs
index 1f11c746a..767ad1961 100644
--- a/uni-stark/src/symbolic_builder.rs
+++ b/uni-stark/src/symbolic_builder.rs
@@ -50,10 +50,10 @@ where
     A: MultiStageAir<SymbolicAirBuilder<F>>,
 {
     let widths: Vec<_> = (0..air.stage_count())
-        .map(|i| air.multi_stage_width(i as u32))
+        .map(|i| air.stage_width(i as u32))
         .collect();
     let challenges: Vec<_> = (0..air.stage_count())
-        .map(|i| air.challenge_count(i as u32))
+        .map(|i| air.stage_challenge_count(i as u32))
         .collect();
     let mut builder = SymbolicAirBuilder::new(
         air.preprocessed_width(),
@@ -188,11 +188,11 @@ impl<F: Field> MultistageAirBuilder for SymbolicAirBuilder<F> {
         &self.public_values[stage]
     }
 
-    fn multi_stage(&self, stage: usize) -> Self::M {
+    fn stage_trace(&self, stage: usize) -> Self::M {
         self.stages[stage].clone()
     }
 
-    fn challenges(&self, stage: usize) -> &[Self::Challenge] {
+    fn stage_challenges(&self, stage: usize) -> &[Self::Challenge] {
         &self.challenges[stage]
     }
 }
diff --git a/uni-stark/src/traits.rs b/uni-stark/src/traits.rs
index 724dc182d..8314b0ceb 100644
--- a/uni-stark/src/traits.rs
+++ b/uni-stark/src/traits.rs
@@ -4,10 +4,10 @@ pub trait MultistageAirBuilder: AirBuilderWithPublicValues {
     type Challenge: Clone + Into<Self::Expr>;
 
     /// Traces from each stage.
-    fn multi_stage(&self, stage: usize) -> Self::M;
+    fn stage_trace(&self, stage: usize) -> Self::M;
 
     /// Challenges from each stage, drawn from the base field
-    fn challenges(&self, stage: usize) -> &[Self::Challenge];
+    fn stage_challenges(&self, stage: usize) -> &[Self::Challenge];
 
     fn stage_public_values(&self, stage: usize) -> &[Self::PublicVar] {
         match stage {
@@ -23,7 +23,7 @@ pub trait MultiStageAir<AB: AirBuilder>: Air<AB> {
     }
 
     /// The number of columns in a given higher-stage trace.
-    fn multi_stage_width(&self, stage: u32) -> usize {
+    fn stage_width(&self, stage: u32) -> usize {
         match stage {
             0 => self.width(),
             _ => unimplemented!(),
@@ -31,7 +31,7 @@ pub trait MultiStageAir<AB: AirBuilder>: Air<AB> {
     }
 
     /// The number of challenges produced at the end of each stage
-    fn challenge_count(&self, _stage: u32) -> usize {
+    fn stage_challenge_count(&self, _stage: u32) -> usize {
         0
     }
 }
diff --git a/uni-stark/src/verifier.rs b/uni-stark/src/verifier.rs
index 41aea03db..420fd3a8d 100644
--- a/uni-stark/src/verifier.rs
+++ b/uni-stark/src/verifier.rs
@@ -74,7 +74,7 @@ where
 
     let air_widths = (0..stages)
         .map(|stage| {
-            <A as MultiStageAir<SymbolicAirBuilder<Val<SC>>>>::multi_stage_width(air, stage as u32)
+            <A as MultiStageAir<SymbolicAirBuilder<Val<SC>>>>::stage_width(air, stage as u32)
         })
         .collect::<Vec<usize>>();
     let air_fixed_width = <A as BaseAir<Val<SC>>>::preprocessed_width(air);