  In this paper, we prove two supercongruences conjectured by Z.-W. Sun via the
Wilf-Zeilberger method. One of them is, for any prime $p>3$, \begin{align*}
\sum_{n=0}^{(p-1)/2}\frac{6n+1}{(-512)^n}\binom{2n}n^3&\equiv
p\left(\frac{-2}p\right)+\frac{p^3}4\left(\frac2p\right)E_{p-3}\pmod{p^4},
\end{align*} where $\left(\frac{\cdot}p\right)$ stands for the Legendre symbol,
and $E_{n}$ is the $n$-th Euler number.
