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The classification of quantum phases of matter is a fundamental topic in the study of quantum many-body systems. A central question in this study is whether or not there is a nonvanishing spectral gap above the ground state energy and, if so, whether or not this gap is stable under sufficiently short-range perturbations. In their seminal work, Affleck, Kennedy Lieb and Tasaki (AKLT) conjectured the existence of a spectral gap for the model they defined on the hexagonal lattice. Significant evidence supporting this long-standing claim was only recently achieved and naturally leads to the question of whether or not this gap is stable. In this talk, we review this and other recent progress on proving gaps for AKLT models and then turn to the question of whether these gaps are stable. One avenue for establishing gap stability pioneered by Bravyi, Hastings and Michalakis is to prove that the finite volume ground states are sufficiently indistinguishable by local observables in the bulk - a property known as Local Topological Quantum Order. We discuss a forthcoming work which uses cluster expansion techniques to prove that the ground states of the AKLT model on the hexagonal lattice and Lieb lattice satisfy LTQO. This talk is based on joint work with Thomas Andrew Jackson and Bruno Nachtergaele.