Is it always true, for a prime $p$, a generator $g$ of $\mathbb{Z}^*_p$ cannot be a quadratic residue modulo $p$?

Question

Let $p$ be a prime, then is it true that a generator $g$ of $\mathbb{Z}^*_p$ cannot be a quadratic residue modulo $p$? And if yes why?

Answer 1

Yes, it is true, for odd $p$.

The point is that a generator $g$ has order $p-1$, which is even.
Yet, if $g= a^2$, then $g^{(p-1)/2} = a^{p-1} = 1$, a contradiction to $p-1$ being the order of $g$.