Impossibility of succinct quantum proofs for collision-freeness

Abstract

We show that any quantum algorithm to decide whether a function f[n] \rightarrow [n] is a permutation or far from a permutation\ must make \Omega( n^{1/3}/w) queries to f, even if the algorithm is given a w-qubit quantum witness in support of f being a permutation. This implies that there exists an oracle A such that {SZK}^{A}\not \subset {QMA}^{A}, answering an eight-year-old open question of the author. \ Indeed, we show that relative to some oracle, {SZK} is not in the counting class {A}_0{PP} defined by Vyalyi. The proof is a fairly simple extension of the quantum lower bound for the collision problem.