Given three nonzero integers, $a,b,c$. If $ab$ is a quadratic residue modulo $c$ and $-ab$ is a quadratic residue modulo $c^{(p-1)/2}$ for a fixed odd prime $p$, what can be said about $a,b,c,$ or $p$?
EDIT: In a related question, if $a,b,c$ are three nonzero integers such that $ab$ is a QR modulo $c^k$ for all $1 \le k \le p-1$ for some fixed odd prime $p$, what can be said about $a,b,c,$ or $p$?