Let  \[ f(x) =
\begin{cases}
-\sqrt[3]x & \text{if } x \geq 0,\\
x^2& \text{if } x <0.
\end{cases}
\]Compute $f(f(f(f(512)))).$
\begin{align*}
f(f(f(f(512))))
&=f(f(f(-8)))\\
&=f(f(64))\\
&=f(-4)\\
&=\boxed{16}.
\end{align*}