On Removing Interaction from Quantum Proofs

Abstract

An important open question in quantum cryptography is the construction of publicly-verifiable NIZKs for QMA. Classically, one can construct NIZKs for NP in the random oracle model (and sometimes in the standard model) by compiling an honest-verifier ZK (HVZK) $\Sigma$-protocol for NP using the Fiat-Shamir transformation. Broadbent and Grilo introduced a quantum analog of a $\Sigma$-protocol (which they call a $\Xi$-protocol) in which the prover's first message is quantum, and show that HVZK $\Xi$-protocols exist for QMA. However, it is not clear how to compile such protocols into NIZKs in the (Q)ROM, because the Fiat-Shamir transformation seems to be incompatible with quantum messages. In this work we give formal evidence that this is indeed the case: we show that if generic"Fiat-Shamir-like"compilers for quantum protocols exist in the QROM (with small completeness and soundness error) then QMA = BQP.