Constant-message protocols with nonclassical verifiers

Determine whether, for every theory satisfying the standing assumptions, a nonclassical verifier exchanging a constant number k≥2 of classical messages has interactive-proof class equal to AM.

Background

For classical verifiers, the paper recalls that any constant number k≥2 of classical messages yields the class AM. The paper establishes that arbitrary theories have class PSPACE when all messages are classical, but it does not characterize the bounded-message subclass when the verifier may use nonclassical internal systems.

A positive answer would already imply BQP⊆AM for the theory QC, because a quantum verifier could ignore the prover. The difficulty identified by the authors is that nonclassical verifier transition probabilities are path sums rather than uniform counts, so the standard set-lower-bound protocol of Goldwasser and Sipser does not immediately extend.

References

With a nonclassical verifier and classical messages, is $GIP(G;k)=AM$ for every theory $G$? {Already for $G=QC$, a positive answer would imply $BQP\subseteqAM$, since the verifier may ignore the prover.} The verifier's transition weights are path sums rather than uniform counts, and the set-lower-bound protocol of Goldwasser and Sipser does not obviously apply to them.

— Beyond IP = PSPACE and QIP = PSPACE: Interactive Proofs in Arbitrary Physical Theories  (2610.00847 - Bharti, 1 Oct 2026) in Section 7, Section “Open questions,” Question “Constant numbers of classical messages” (label q:constant)