Before revision: 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.
After revision: We use the recording queries technique of Zhandry [ Zha19 ] \rightarrow ] . Finally, we use the latter result to obtain the first quantum time-space tradeoff that is not based on a reduction to K-Search.