Simulating Time with Square-Root Space

DWe show that for all functions.. (..) =.., every multitape Turing machine running in time.. can be simulated in space only.. ( v... log..). This is a substantial improvement over Hopcroft, Paul, and Valiant's simulation of time.. in.. (.. /log..) space from 50 years ago [FOCS 1975, JACM 1977]. Among other results, our simulation implies that bounded fan-in circuits of size.. can be evaluated on any input in only v.. center dot........ (log..) space, and that there are explicit problems solvable in.. (..) space which require at least.. 2-.. time on every multitape Turing machine for all.. > 0, thereby making a little progress on the P versus 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].