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Given $m$ matrices of size $n\times n$ each of which is promised to be a permutation is it in $\mathit{quasiAC}^0$ or $\mathit{AC}^0$ to multiply the permutations where

  1. $m=\mathit{poly}(n)$
  2. $m=\mathit{poly}(\log n)$ which means $m=O(\log^k n)$ where $k\in\mathbb N_{>1}$
  3. $m=O(\log n)$?

Given $\mathit{poly}(n)$ inputs it is clearly in $\mathit{AC}^0$ to test the promise every input matrix is a permutation.

On other hand general iterated matrix multiplication is is $\mathit{NC}^2$.

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Ben Rossman showed that any unbounded fan-in depth $d$ circuit for your problem has size at least $n^{\Omega(m^{1/2d})}$. Conversely, a simple recursive construction gives an unbounded fan-in depth $d$ formula of size $n^{O(m^{1/d})}$.

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    $\begingroup$ I trust that you can do the arithmetic on your own. $\endgroup$ Apr 2, 2021 at 21:22
  • $\begingroup$ The bound stated is incorrect or else it separate NC1 and AC1. I think m is restricted. $\endgroup$
    – Turbo
    Jun 8, 2021 at 0:21
  • $\begingroup$ Do you know the correct reference (I see R'08 R'14 etc but no right reference)? Are you certain $m$ is not bounded by $\log n$ or $\log\log n$ or anything? $\endgroup$
    – Turbo
    Jun 8, 2021 at 6:19
  • $\begingroup$ I’m not sure it has been published yet. In the slides there are some limits on $d$. $\endgroup$ Jun 8, 2021 at 6:49
  • $\begingroup$ Yes unlikely these tricks can prove lower bounds. $\endgroup$
    – Turbo
    Jun 8, 2021 at 8:03

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