  We study discretizations of fractional fully nonlinear equations by powers of
discrete Laplacians. Our problems are parabolic and of order $\sigma\in(0,2)$
since they involve fractional Laplace operators $(-\Delta)^{\sigma/2}$. They
arise e.g.~in control and game theory as dynamic programming equations, and
solutions are non-smooth in general and should be interpreted as viscosity
solutions. Our approximations are realized as finite-difference quadrature
approximations and are 2nd order accurate for all values of $\sigma$. The
accuracy of previous approximations depend on $\sigma$ and are worse when
$\sigma$ is close to $2$. We show that the schemes are monotone, consistent,
$L^\infty$-stable, and convergent using a priori estimates, viscosity solutions
theory, and the method of half-relaxed limits. We present several numerical
examples.
