Space complexity for multiplying $m$ matrices Suppose you have $m$ $n$ by $n$ matrices $M_1,M_2,\dotsc,M_m$, and you want to calculate their product $\prod_{i=1}^{m} M_i$. 

The naive method use $m \cdot poly(n)$ times but needs $poly(n)$ memory. I am wondering can there be any algorithm with $polylog(n,m)$ memory but still $m \cdot poly(n)$ or $poly(n,m)$ running time. It sounds like an old problem but I couldn't find any reference.

Note that using the divide and conquer technique (like in Savitch's theorem), we can achieve $polylog(n,m)$ memory and $m \cdot n^{O(\log m)}$ running time.
Letting m be [number of vertices minus 1] and letting each matrix be

[the adjacency matrix of the result of giving the vertex t a loop]

gives a reduction from st-connectivity to your problem, so no such algorithm is known.