We show that for all functions $t(n) \geq n$, 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, ... more >>>

We prove that relative to a random oracle answering $O(\log n)$-bit queries, there exists a function computable in $O(n)$ time by a random-access machine (RAM) but requiring $n^2/polylog(n)$ time by any multitape Turing machine. This provides strong evidence that simulating RAMs on multitape Turing machines inherently incurs a nearly quadratic ... more >>>