Some determinants involving quadratic residues modulo primes

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