$g$ generates $\Bbb Z_p$ and $g$ is not a quadratic resiude.
Supposing $g^{qz}=h\pmod p$ where $p=2q+1$ is prime and $q$ is prime and $z$ is even then $h=1\bmod p$. So we cannot find what $z$ was.
Supposing $g^{qz}=h\pmod p$ where $p=2q+1$ is prime and $q$ is prime and $z$ is odd then can we find $z$ in polynomial time (or are we still stuck with sieve techniques)?