Prove that if $a$ is a quadratic residue of $p$ then $a$ is not a primitive root of $p$.
If $a$ is a quadratic residue of $p$ then we know $p \not \mid a$, and so by Euler's Criterion,
$$ a^{\frac{p-1}{2}} \equiv 1 \pmod p $$
This means $a$ is not a primitive root of $p$, because that would require the smallest index $j$ of $a^j \equiv 1 \pmod p$ to be $j=p-1$.