Given $k \geq 1$, what is the minimal $n$ (if it exists) such that there're matrices $A_1, \cdots A_k \in M_n(\mathbb{C}) $ satisfying $\forall i,j \quad A_i A_j = - A_j A_i$ and $A_1 \times A_2 \cdots \times A_k \neq 0$ ? In particular each $A_i ^2$ must be zero.

Motivation : make Ramyslov's elementary proof of the Amitsur–Levitzki identity look even more elementary.