  We study the computational complexity of finding a solution for the
straight-cut and square-cut pizza sharing problems. We show that computing an
$\varepsilon$-approximate solution is PPA-complete for both problems, while
finding an exact solution for the square-cut problem is FIXP-hard and in BU.
Our PPA-hardness results apply for any $\varepsilon < 1/5$, even when all mass
distributions consist of non-overlapping axis-aligned rectangles or when they
are point sets, and our FIXP-hardness result applies even when all mass
distributions are unions of squares and right-angled triangles. We also prove
that the decision variants of both approximate problems are NP-complete, while
it is ETR-complete for the exact version of square-cut pizza sharing.
