Characterizing Quantum Advantage for Generalizations of the Boolean Hidden Matching Problem

We study the one-way communication complexity of the $f$-Boolean Hidden Partition problem. Here, Alice is given an $n$-bit string and Bob is given $Ω(n)$ disjoint blocks of its indices together with a string of labels. Under the promise that the evaluations of $f$ on these blocks either agree with all of the labels or disagree with all of them, Bob must determine which is the case using a single message from Alice. This problem generalizes the Boolean Hidden Matching and Hidden Hypermatching problems, which capture the special case where $f$ is the parity function. We establish classical and quantum communication bounds for $f$-Boolean Hidden Partition in terms of the sign degree $d$ of $f$, proving a conjecture of Doriguello and Montanaro (TQC 2020). Their prior work gave logarithmic communication upper bounds for classical protocols when $d \le 1$ and quantum protocols when $d \le 2$, as well as polynomial lower bounds for certain structured functions with larger sign degree. We show that for every $f$ of sign degree $d \ge 2$, the randomized classical communication complexity of this problem is $Θ(n^{1-1/d})$ while its quantum communication complexity lies between $Ω(n^{1-2/d})$ and $\bO{n^{1-1/\lceil d/2 \rceil} \log n}$. This completely characterizes the functions $f$ admitting polynomial quantum advantage as those for which $d \ge 2$, with a new infinite family of exponential separations given by the case $d = 2$.