Quantum Time-Space Tradeoffs by Recording Queries

We use the recording queries technique of Zhandry [Zha19] to prove lower bounds in the exponentially small success probability regime, with applications to time-space tradeoffs. We first extend the recording technique to the case of non-uniform input distributions and we describe a new simple framework for using it. Then, as an application, we prove strong direct product theorems for $K$-Search under a natural product distribution not considered in previous works, and for finding $K$ distinct collisions in a uniform random function. Finally, we use the latter result to obtain the first quantum time-space tradeoff that is not based on a reduction to $K$-Search. Namely, we demonstrate that any $T$-query algorithm using $S$ qubits of memory must satisfy a tradeoff of $T^3 S \geq \Omega(N^4)$ for finding $\Theta(N)$ collisions in a random function. We conjecture that this result can be improved to $T^2 S \geq \Omega(N^3)$, and we show that it would imply a $T^2 S \geq \tilde{\Omega}(N^2)$ tradeoff for Element Distinctness.