On the Need for (Quantum) Memory with Short Outputs

TL;DR

This paper demonstrates a fundamental separation in computational complexity between bounded and unbounded memory for short-output problems, using quantum and classical models, and introduces a novel technique for analyzing time-space tradeoffs.

Contribution

It establishes the first separation between bounded and unbounded space for short-output problems and introduces a new 'two-oracle recording' technique for analyzing time-space tradeoffs.

Findings

Optimal query complexity requires exponential memory for the nested collision finding problem.

The 'two-oracle recording' technique effectively reduces short-output problems to long-output ones.

The results hold in both classical and quantum computational models.

Abstract

In this work, we establish the first separation between computation with bounded and unbounded space, for problems with short outputs (i.e., working memory can be exponentially larger than output size), both in the classical and the quantum setting. Towards that, we introduce a problem called nested collision finding, and show that optimal query complexity can not be achieved without exponential memory. Our result is based on a novel ``two-oracle recording'' technique, where one oracle ``records'' the computation's long outputs under the other oracle, effectively reducing the time-space trade-off for short-output problems to that of long-output problems. We believe this technique will be of independent interest for establishing time-space tradeoffs in other short-output settings.