  So far, it has been proven that if $G$ is an abelian group , then the
diameter of $G^n$ with respect to any generating set is $O(n)$; and if $G$ is
nilpotent, symmetric or dihedral, then there exists a generating set of minimum
size, for which the diameter of $G^n$ is $O(n)$ \cite{Karimi:2017}. In
\cite{Dona:2022} it has been proven that if $G$ is a non-abelian simple group,
then the diameter of $G^n$ with respect to any generating set is $O(n^3)$. In
this paper we estimate the diameter of direct power of alternating groups $A_n$
for $n \geq 4$, i.e. a class of non-abelian simple groups. We show that there
exist a generating set of minimum size for $A_4^n$, for which the diameter of
$A_4^n$ is $O(n)$. For $n \geq 5$, we show that there exists a generating set
of minimum size for $A_n^2$, for which the diameter of $A_n^2$ is at most
$O(ne^{(c+1) (\log \,n)^4 \log \log n})$ , for an absolute constant $c >0$.
Finally for $ 1\leq n \leq 8 $, we provide generating sets of size two for
$A_5^n$ and we show that the diameter of $A_5^n$ with respect to those
generating sets is $O(n)$. These results are more pieces of evidence for a
conjecture which has been presented in \cite{Karimithesis:2015} in 2015.
