Upper Bound on the entries of $A^n$, where $A \in M_d(\mathbb C)$ and $n\in \mathbb N$
The best upper bound is $d^{n-1}M^n$: \begin{aligned} |(A^n)_{ij}| &=\left|\sum_{(k_1,k_2,\ldots,k_{n-1})\in[d]^{n-1}} a_{ik_1}a_{k_1k_2}\cdots a_{k_{n-1}j}\right|\\ &\le\sum_{(k_1,k_2,\ldots,k_{n-1})\in[d]^{n-1}} |a_{ik_1}a_{k_1k_2}\cdots a_{k_{n-1}j}|\\ &\le\sum_{(k_1,k_2,\ldots,k_{n-1})\in[d]^{n-1}}M^n\\ &=d^{n-1}M^n. \end{aligned} This bound is sharp. It is attained when every element of $A$ is equal to $M$.