  We consider the complex cut polytope: the convex hull of Hermitian rank 1
matrices $xx^{\mathrm{H}}$, where the elements of $x \in \mathbb{C}^n$ are
$m$th unit roots. These polytopes have applications in ${\text{MAX-3-CUT}}$,
digital communication technology, angular synchronization and more generally,
complex quadratic programming. For ${m=2}$, the complex cut polytope
corresponds to the well-known cut polytope. We generalize valid cuts for this
polytope to cuts for any complex cut polytope with finite $m>2$ and provide a
framework to compare them. Further, we consider a second semidefinite lifting
of the complex cut polytope for $m=\infty$. This lifting is proven to be
equivalent to other complex Lasserre-type liftings of the same order proposed
in the literature, while being of smaller size. We also prove that a second
semidefinite lifting of the complex cut polytope for $n = m=3$ is exact. Our
theoretical findings are supported by numerical experiments on various
optimization problems.
