  We use low-depth quantum circuits, a specific type of tensor networks, to
classify two-dimensional symmetry-protected topological many-body localized
phases. For (anti-)unitary on-site symmetries we show that the (generalized)
third cohomology class of the symmetry group is a topological invariant;
however our approach leaves room for the existence of additional topological
indices. We argue that our classification applies to quasi-periodic systems in
two dimensions and systems with true random disorder within times which scale
superexponentially with the inverse interaction strength. Our technique might
be adapted to supply arguments suggesting the same classification for
two-dimensional symmetry-protected topological ground states with a rigorous
proof.
