Simulating Time with Square-Root Space

We show that for all functions t ( n ) ≥ n , every multitape Turing machine running in time t can be simulated in space only $O(\sqrt {t \log t})$ . This is a substantial improvement over Hopcroft, Paul, and Valiant’s simulation of time t in O ( t /log t ) space from 50 years ago [FOCS 1975, JACM 1977]. Among other results, our simulation implies that bounded fan-in circuits of size s can be evaluated on any input in only $\sqrt {s} \cdot \text{poly}(\log s)$ space, and that there are explicit problems solvable in O ( n ) space which require n 2 − ε time on a multitape Turing machine for all ε > 0, thereby making a little progress on the ${\sf P}$ versus ${\sf PSPACE}$ problem. Our simulation reduces the problem of simulating time-bounded multitape Turing machines to a series of implicitly-defined Tree Evaluation instances with nice parameters, leveraging the remarkable space-efficient algorithm for Tree Evaluation recently found by Cook and Mertz [STOC 2024].