  We study the computational complexity of computing solutions for the
square-cut pizza sharing problem. In this problem, we have $n$ mass
distributions in the plane, and the task is to find a path that uses horizontal
and vertical segments that splits each of the masses in half while making at
most $n-1$ turns. We show that finding an approximate solution to this problem
is PPA-complete, while finding an exact solution is FIXP-hard and in BU. Our
PPA-hardness result applies even when all mass distributions are unions of
non-overlapping squares, and our FIXP-hardness result applies even when all
mass distributions are unions of weighted squares and right-angled triangles.
When the path is restricted to have at most $n-2$ turns, we show that the
approximate problem becomes NP-complete, and the exact problem becomes
ETR-complete.
