  The $k^{\text{th}}$ power of a graph $G=(V,E)$, $G^k$, is the graph whose
vertex set is $V$ and in which two distinct vertices are adjacent if and only
if their distance in $G$ is at most $k$. This article proves various eigenvalue
bounds for the independence number and chromatic number of $G^k$ which purely
depend on the spectrum of $G$, together with a method to optimize them. Our
bounds for the $k$-independence number also work for its quantum counterpart,
which is not known to be a computable parameter in general, thus justifying the
use of integer programming to optimize them. Some of the bounds previously
known in the literature follow as a corollary of our main results. Infinite
families of graphs where the bounds are sharp are presented as well.
