  In this paper we evaluate several determinants involving quadratic residues
modulo primes. For example, for any prime $p>3$ with $p\equiv3\pmod4$ and
$a,b\in\mathbb Z$ with $p\nmid ab$, we prove that $$\det\left[
1+\tan\pi\frac{aj^2+bk^2}p \right]_{1\le j,k \le \frac{p-1}2} =
\begin{cases}-2^{(p-1)/2}p^{(p-3)/4}&\text{if}\ (\frac{ab}p)=1,
\\p^{(p-3)/4}&\text{if}\ (\frac{ab}p)=-1, \end{cases}$$ where $(\frac{\cdot}p)$
denotes the Legendre symbol. We also pose some conjectures for further
research.
