  We prove sufficient conditions for Topological Quantum Order at both zero and
finite temperatures. The crux of the proof hinges on the existence of
low-dimensional Gauge-Like Symmetries (that notably extend and differ from
standard local gauge symmetries) and their associated defects, thus providing a
unifying framework based on a symmetry principle. These symmetries may be
actual invariances of the system, or may emerge in the low-energy sector.
Prominent examples of Topological Quantum Order display Gauge-Like Symmetries.
New systems exhibiting such symmetries include Hamiltonians depicting
orbital-dependent spin exchange and Jahn-Teller effects in transition metal
orbital compounds, short-range frustrated Klein spin models, and p+ip
superconducting arrays. We analyze the physical consequences of Gauge-Like
Symmetries (including topological terms and charges), discuss associated
braiding, and show the insufficiency of the energy spectrum, topological
entanglement entropy, maximal string correlators, and fractionalization in
establishing Topological Quantum Order. General symmetry considerations
illustrate that not withstanding spectral gaps, thermal fluctuations may impose
restrictions on certain suggested quantum computing schemes and lead to
"thermal fragility". Our results allow us to go beyond standard topological
field theories and engineer systems with Topological Quantum Order.
