Reduce the update complexity of the ARHM separation

Determine whether the Atomized Refreshable Hidden Matching construction can achieve the same quantum random-order versus classical and adversarial-order space separation using fewer than O(Nr^2) coordinate updates.

Background

The Atomized Refreshable Hidden Matching (ARHM) construction uses O(Nr2) repeated coordinate updates. These repetitions serve two purposes: controlling the phase-approximation error of the replenished Hidden Matching states and ensuring that sufficiently many completed replacement registers are available when requests arrive.

The paper explicitly asks whether this update overhead is necessary for obtaining the same space separation. Resolving the question would clarify the quantitative cost of replenishment rather than merely its qualitative possibility.

References

Our construction uses $\bT{Nr2}$ coordinate updates to control the phase error and supply enough completed registers. Can fewer updates give the same space separation?

— Random Order in Quantum Streaming: Replenishment and Robust Lower Bounds  (2610.06727 - Voronova, 5 Oct 2026) in Section 4, paragraph "Open questions" following the proof of Theorem 4.1