On the Need for (Quantum) Memory with Short Outputs

2026-02-27

AI Summary

This work investigates whether bounded and unbounded space computation exhibit an inherent separation when the output length is significantly shorter than the working memory. To address this question, the authors introduce the “nested collision finding” task and propose a novel technique called “double-oracle recording,” which reduces the short-output setting to a long-output one for analysis. Leveraging query complexity and time–space tradeoff frameworks in both classical and quantum computational models, they establish—for the first time in the short-output regime—that achieving optimal query complexity is impossible without exponential memory. This result demonstrates a provable separation between bounded- and unbounded-memory computation and yields new lower bounds in computational complexity theory.

Technology Category

• Search and Optimization: Learning to Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Quantum Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Security and Privacy: Large-scale security measurements

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.

Problem

• time-space tradeoff
• short-output problems
• memory complexity
• collision finding
• quantum computation

Innovation

• time-space tradeoff
• quantum memory
• nested collision finding
• two-oracle recording
• short-output computation