  The extension complexity of a polytope measures its amenability to succinct
representations via lifts. There are several versions of extension complexity,
including linear, real semidefinite, and complex semidefinite. We focus on the
last of these, for which the least is known, and in particular on understanding
which polytopes are complex psd-minimal. We prove the existence of an
obstruction to complex psd-minimality which is efficiently computable via
lattice membership problems. Using this tool, we complete the classification of
complex psd-minimal polygons (geometrically as well as combinatorially). In
dimension three we exhibit several new examples of complex psd-minimal
polytopes and apply our obstruction to rule out many others.
