Time-space tradeoff has been studied in a variety of models, such as Turing machines, branching programs, and finite automata, etc. While communication complexity as a technique has been applied to study finite automata, it seems it has not been used to study time-space tradeoffs of finite automata. We design a new technique showing that separations of query complexity can be lifted, via communication complexity, to separations of time-space complexity of two-way finite automata. As an application, one of our main results exhibits the first example of a language $L$ such that the time-space complexity of two-way probabilistic finite automata with a bounded error (2PFA) is $\widetilde{\Omega}(n^2)$, while of exact two-way quantum finite automata with classical states (2QCFA) is $\widetilde{O}(n^{5/3})$, that is, we demonstrate for the first time that exact quantum computing has an advantage in time-space complexity comparing to classical computing.

翻译：时间-空间权衡已在多种模型中得到研究，例如图灵机、分支程序和有限自动机等。尽管通信复杂度作为一种技术已被应用于研究有限自动机，但它似乎尚未被用于研究有限自动机的时间-空间权衡。我们设计了一种新技术，表明查询复杂度的分离可以通过通信复杂度提升为双向有限自动机的时间-空间复杂度的分离。作为应用，我们的主要结果之一首次给出了一个语言$L$的示例，使得有界误差双向概率有限自动机（2PFA）的时间-空间复杂度为$\widetilde{\Omega}(n^2)$，而精确双向经典态量子有限自动机（2QCFA）的时间-空间复杂度为$\widetilde{O}(n^{5/3})$。也就是说，我们首次证明，与经典计算相比，精确量子计算在时间-空间复杂度方面具有优势。