2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/147465I discuss the concept of quasi-state decompositions for ground states and equilibrium states of quantum spin systems. Some recent results on the ground states of a class of one-dimensional quantum spin models are summarized and new work in progress is presented. I also outline some challenging open problems and conjectures.18 pages, Plain TeX, 5 figures available upon requestCondensed Mattertext