Optimal Separations Between Randomized and Quantum Query Complexity under Permutation Symmetry

arXiv:2611.08919v1 Announce Type: new Abstract: A function $f:\cD\subseteq A^N\to\{0,1\}$ is \emph{permutation-invariant} if every permutation of its input positions preserves both its domain and its value. We study how this symmetry limits quantum speedups. For $1/6\leq\varepsilon<1/2$, we prove that every such function over an arbitrary finite alphabet $A$ satisfies $R_\varepsilon(f)=O(Q_\varepsilon(f)^2)$, where $R_\varepsilon$ and $Q_\varepsilon$ denote randomized and quantum query complexities with error at most $\varepsilon$. This improves the previous cubic bound (ITCS 2019). We further establish bounds for graph and hypergraph properties, for which vertex relabeling preserves both the promise and the answer.