Tonelli–Shanks algorithm needs quadratic nonresidue $n$ modulo prime $p$. If the extended Riemann hypothesis is true, then the first quadratic nonresidue $n_p$ is always less than $3(\log p)^2/2$ (Wedeniwski 2001) for $ p>3$. Under this assumption Tonelli–Shanks algorithm becomes polynomial. Probabilistic heuristics (presuming that each non-square integer has a 50-50 chance of being a quadratic residue) suggests that $n_p$ should have size $O( \log p )$. The best unconditional estimate (Burgess) is $n_p\ll p^\theta$ for any $\theta >1/4\sqrt e$.
See also extended discussion at The least quadratic nonresidue, and the square root barrier.