  Let $p$ be a large odd prime, let $x=(\log p)^{1+\varepsilon}$ and let
$q\ll\log\log p$ be an integer, where $\varepsilon>0$ is a small number. This
note proves the existence of small prime quadratic residues and prime quadratic
nonresidues in the arithmetic progression $a+qm\ll x$, with relatively prime
$1\leq a<q$, unconditionally. The same results are generalized to small prime
$k$th power residues and nonresidues, where $k\mid p-1$ and $k\ll\log\log p$.
