  We introduce deterministic state-transformation protocols between many-body
quantum states which can be implemented by low-depth Quantum Circuits (QC)
followed by Local Operations and Classical Communication (LOCC). We show that
this gives rise to a classification of phases in which topologically-ordered
states or other paradigmatic entangled states become trivial. We also
investigate how the set of unitary operations is enhanced by LOCC in this
scenario, allowing one to perform certain large-depth QC in terms of low-depth
ones.
