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.

翻译：在本工作中，我们首次在经典与量子设定下，针对短输出问题（即工作存储可比输出规模呈指数级更大）建立了有界空间与无界空间的计算分离结果。为此，我们引入了一个名为"嵌套碰撞查找"的问题，并证明若无指数级存储，则无法实现最优查询复杂度。我们的研究成果基于一种新颖的"双预言机记录"技术，其中一个预言机"记录"另一预言机下计算的长输出，从而将短输出问题的时间-空间权衡有效转化为长输出问题的时间-空间权衡。我们相信该技术将在其他短输出设定中建立时间-空间权衡时具有独立价值。