List Distinguishing Number of Power of Hypercube and Cartesian Powers of a Graph

The list distinguishing number $D_l\left(G\right)$ measures how many colors each vertex’s allowed list must contain to guarantee a coloring that breaks every nontrivial symmetry. We show that list restrictions do not increase the distinguishing number of Cartesian powers $G^r$ for connected Cartesian-prime graphs and $r\ge 3$, and determine $D_l\left(Q_n^{\left[p\right]}\right)$ for powers of hypercubes with $n\ge 7$ and $p\ge 1$.