  Let $e_{1},\dots, e_{k}$ be complex $n\times n$ matrices such that
$e_{i}e_{j}=-e_{j}e_{i}$ whenever $i\not=j$. We conjecture that
$\hbox{rk}(e_{1}^{2})+\hbox{rk}(e_{2}^{2})+\cdots+\hbox{rk}(e_{k}^{2})\leq
O(n\log n)$, and prove some results in this direction.
