Structured Quantum Hardness from Unstructured Assumptions

arXiv:2611.07413v1 Announce Type: new Abstract: Whether useful structured hardness can arise from unstructured assumptions is a fundamental question in complexity and cryptography. The longstanding Bennett-Gill question asks whether $\mathsf{NP}\cap\mathsf{coNP}$ remains hard relative to a random oracle; [GIKKLS, STOC 2023] recently gave a conditional affirmative answer in the classical setting. We establish the quantum analogue: if $\mathsf{UP}\not\subseteq\mathsf{BQP}$, then $\mathsf{NP}^{O}\cap\mathsf{coNP}^{O}\not\subseteq\mathsf{BQP}^{O}$ with probability $1$ over a random oracle $O$.