  We study several aspects of the realization of global symmetries in highly
entangled phases of quantum matter. Examples include gapped topological ordered
phases, gapless quantum spin liquids and non-fermi liquid phases. An insightful
window into such phases is provided by recent developments in the theory of
short ranged entangled Symmetry Protected Topological (SPT) phases . First they
generate useful no-go constraints on how global symmetry may be implemented in
a highly entangled phase. Possible symmetry implementation in gapped
topological phases and some proposed gapless spin/bose liquids are examined in
this light. We show that some previously proposed spin liquid states for 2d
quantum magnets do not in fact have consistent symmetry implementation unless
they occur as the surface of a 3d SPT phase. A second SPT-based insight into
highly entangled states is the development of a view point of such states as
SPT phases of one of the emergent excitations. We describe this in the specific
context of time reversal symmetric 3d U(1) quantum spin liquids with an
emergent photon. Different such spin liquids are shown to be equivalent to
different SPT insulating phases of the emergent monopole excitation of such
phases. The highly entangled states also in turn enrich our understanding of
SPT phases. We use the insights obtained from our results to provide an
explicit construction of bosonic SPT phases in 3d in a system of coupled
layers. This includes construction of a time reversal symmetric SPT state that
is not currently part of the cohomology classification of such states.
