Proof of some conjectures involving quadratic residues

We confirm several conjectures of Sun involving quadratic residues modulo odd primes. For any prime $ p\equiv 1\ ({\rm{mod}}\ 4) $ and integer $ a\not\equiv0\ ({\rm{mod}}\ p) $, we prove that

$ (-1)^{|\{1 \leq k<\frac p4:\ (\frac kp) = -1\}|}\prod\limits_{1 \leq j<k \leq (p-1)/2}(e^{2\pi iaj^2/p}+e^{2\pi iak^2/p}) \\ = \begin{cases}1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{if}\ p\equiv1\ ({\rm{mod}}\ 8), \\ \left(\frac ap\right)\varepsilon_p^{-(\frac ap)h(p)}\ \ \text{if}\ p\equiv5\ ({\rm{mod}}\ 8), \end{cases} $

and that

$ \begin{array}{*{35}{l}}\left|\left\{(j, k):\ 1 \leq j<k \leq \frac{p-1}2\ \ \{aj^2\}_p>\{ak^2\}_p\right\}\right| \\ +\left|\left\{(j, k):\ 1 \leq j<k \leq \frac{p-1}2\ \ \{ak^2-aj^2\}_p>\frac p2\right\}\right| \\ \equiv \left|\left\{1 \leq k<\frac p4:\ \left(\frac kp\right) = \left(\frac ap\right)\right\}\right|\ ({\rm{mod}}\ 2), \end{array}$

where $ (\frac{a}p) $ is the Legendre symbol, $ \varepsilon_p $ and $ h(p) $ are the fundamental unit and the class number of the real quadratic field $ \mathbb Q(\sqrt p) $ respectively, and $ \{x\}_p $ is the least nonnegative residue of an integer $ x $ modulo $ p $. Also, for any prime $ p\equiv3\ ({\rm{mod}}\ 4) $ and $ {\delta} = 1, 2 $, we determine

$ (-1)^{\left|\left\{(j, k): \ 1 \leq j<k \leq (p-1)/2\ \text{and}\ \{{\delta} T_j\}_p>\{{\delta} T_k\}_p\right\}\right|}, $

where $ T_m $ denotes the triangular number $ m(m+1)/2 $.