  We discuss systems which have some, but not all of the hallmarks of
topological phases. These systems' topological character is not fully captured
by a local order parameter, but they are also not fully described at low
energies by topological quantum field theories. For such systems, we formulate
the concepts of quasi-topological phases (to be contrasted with true
topological phases), and symmetry-protected quasi-topological phases. We
describe examples of systems in each class and discuss the implications for
topological protection of information and operations. We explain why
topological phases and quasi-topological phases have greater stability than is
sometimes appreciated. In the examples that we discuss, we focus on Ising-type
(a.k.a. Majorana) systems particularly relevant to recent theoretical advances
and experimental efforts.
