Lifting query complexity to time-space complexity for two-way finite automata

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 $\tilde{\Omega }\left(n^2\right)$, while of exact two-way quantum finite automata with classical states (2QCFA) is $\tilde{O}\left(n^{5/3}\right)$, that is, we demonstrate for the first time that exact quantum computing has an advantage in time-space complexity comparing to classical computing.