Extend simultaneous Hidden Matching lower bounds to the square-root endpoint

Prove a linear communication lower bound for simultaneous Multiple Hidden Matching requests when the number of requests is $m=\floor{\sqrt N}$, matching the sequential-request lower bound.

Background

The paper proves a linear lower bound for simultaneous requests only when m≤N1/2−δm\le N^{1/2-\delta} for a fixed positive δ\delta, whereas its sequential-request argument reaches the square-root scale. At the endpoint, the paper obtains only an Ω~(N)\widetilde{\Omega}(\sqrt N)-type bound with a logarithmic loss.

The unresolved issue is whether joint decoding of all requests fundamentally permits a smaller representation at m=Nm=\sqrt N, or whether the sequential linear dependence also holds for simultaneous revelation.

References

For simultaneous requests, our linear lower bound holds when $m\le N{1/2-\delta}$ for fixed $\delta>0$. Can it reach $m=\floor{\sqrt N}$, as the sequential bound does?

— Random Order in Quantum Streaming: Replenishment and Robust Lower Bounds  (2610.06727 - Voronova, 5 Oct 2026) in Section 5, paragraph "Open questions"

Even where our lower bound is linear, sending separate phase states uses $O{m\log N}$ qubits. What is the optimal joint dependence on $N$ and $m$?

— Random Order in Quantum Streaming: Replenishment and Robust Lower Bounds  (2610.06727 - Voronova, 5 Oct 2026) in Section 5, paragraph "Open questions"