Establish a natural replenishment-based random-order separation

Establish a comparable separation between random-order and adversarial-order streaming space for a natural problem using quantum replenishment, rather than for the artificial ARHM problem.

Background

ARHM demonstrates that continued access to repeated classical data can replenish consumable quantum states and produce a strong random-order versus adversarial-order separation, but ARHM is explicitly designed as an artificial problem.

The triangle-counting application concerns a natural problem and improves an existing algorithm, yet it does not establish a corresponding random-versus-adversarial separation. The paper identifies proving such a separation for a natural problem as unresolved.