Consider the Decision Diffie-Hellman (DDH) over $QR_n$ (the quadratic residue group over $n=pq$ where $p$ and $q$ are safe primes).According to Boneh's paper, DDH should be hard over $QR_n$ (https://link.springer.com/chapter/10.1007/BFb0054851):
[DDH] Given three randomly sampled $g^x, g^y, g^z$ it is hard to tell if $z = x*y$.
I'm wondering: if given an extra $x^2$ $mod$ $n$, is this problem still hard over $QR_n$? (The intuition is that computing the root of $x^2$ is hard, so intuitively this square should provide no additional information for solving DDH. The exceptions might be it leaks Jacobi/Legendre symbols, but the random pick of $z$ can be fixed correspondingly?).