Tightness of the mixed time–space term for one-way states

Determine whether the mixed term \(\sqrt{ST}/N\) in the timespace security bound for recovering the key from a random binary phase state is tight, and establish whether barriers analogous to those in the classical preprocessing setting obstruct identifying the correct dependence.

Background

The paper proves that a quantum adversary with $S$ qubits of advice and $T$ random-oracle queries recovers the key of a random binary phase state with probability at most $O((T^2+1+\sqrt{S(T+1)})/N)$ in the regime $K=N$. The term involving $\sqrt{S(T+1)}$ captures the interaction between preprocessing space and online queries, while the $T^2$ term corresponds to the online-query contribution. The authors note that the bound is tight when either the advice is absent or the adversary makes no online queries, but the tightness of the mixed term remains unresolved. They also connect this issue to classical timespace tradeoffs, where sharpening analogous bounds would have implications for circuit lower bounds.