How can I efficiently find all the Quadratic residue of some prime number $x$?
$$\{y ~\vert~ x^2 \equiv y \pmod{p}\}$$
In wiki they speak about some thing called lifting, by I am not sure how to use it.
If the modulus n is a prime power n = pe, a solution may be found modulo p and "lifted" to a solution modulo n using Hensel's lemma or an algorithm of Gauss
If you want to find all of them, then simply computing $y^2 \bmod x$ for all $y$ between $0$ and $\frac{x-1}{2}$ will produce all of them, in some order, at a cost of one modular squaring per number being output. It's difficult to imagine how one could hope for a faster method than that.