First, two different representations and inner and outer nonlinear approximations of the (real) max-cut polytope are discussed. Applications from digital communication lead to a complex analogue of the max-cut polytope, the complex "unit modulus lifting". The definition of the complex unit modulus lifting is rather similar to its real counterpart, but it does not allow a generalization of triangle inequalities. Instead, for $n=3$, the semidefinite approximation is exact. Some relations to set-completely-positive representations in the real and in the complex case close the presentation.
