On the degree of polynomials that approximate symmetric Boolean functions (preliminary version)
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Quantum cryptanalysis of hash and claw-free functions
ACM SIGACT News
Strengths and Weaknesses of Quantum Computing
SIAM Journal on Computing
Quantum lower bound for the collision problem
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Complexity measures and decision tree complexity: a survey
Theoretical Computer Science - Complexity and logic
Quantum lower bounds for the collision and the element distinctness problems
Journal of the ACM (JACM)
Error-bounded probabilistic computations between MA and AM
Journal of Computer and System Sciences
Quantum versus Classical Proofs and Advice
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
Computational Complexity
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We show that any quantum algorithm to decide whether a function f : [n] → [n] is a permutation or far from a permutation must make Ω (n1/3/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 SZKA ⊄ QMAA, 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 A0PP defined by Vyalyi. The proof is a fairly simple extension of the quantum lower bound for the collision problem.