  In this paper, we prove a quantitative version of the statement that every
nonempty finite subset of $\mathbb{N}^+$ is a set of quadratic residues for
infinitely many primes of the form $[n^c]$ with $1\leqslant c\leqslant243/205$.
Correspondingly, we can obtain a similar result for the case of quadratic
non-residues under reasonable assumptions. These results generalize the
previous ones obtained by S. Wright in certain aspects.
