Proof of two congruence conjectures of Z.-W. Sun

In this paper, we mainly prove two congruence conjecture of Z.-W. Sun. Let $p\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$ be a prime. Then $$\sum _{k=0}^{p-1}\frac{{\left(\genfrac{}{}{0px}{}{2k}{k}\right)}^2}{8^k}\equiv -\sum _{k=0}^{p-1}\frac{{\left(\genfrac{}{}{0px}{}{2k}{k}\right)}^2}{(-16{)}^k}(\mathrm{m}\mathrm{o}\mathrm{d}{p}^3)\mathrm{.}$$ And for any odd prime $p$, if $p=x^2+y^2$ with $4\mathrm{\mid }x-1,2\mathrm{\mid }y$, then $$\sum _{k=0}^{p-1}\frac{(k+1){\left(\genfrac{}{}{0px}{}{2k}{k}\right)}^2}{8^k}+\sum _{k=0}^{(p-1)\mathrm{/}2}\frac{(2k+1){\left(\genfrac{}{}{0px}{}{2k}{k}\right)}^2}{(-16{)}^k}\equiv 2\left(\frac{2}{p}\right)x(\mathrm{m}\mathrm{o}\mathrm{d}{p}^3)\mathrm{.}$$