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We explore generalizations to arbitrary <jats:italic>D<\/jats:italic> of SU(2)-spin coherent states and their adaptation to parity (multi-component Schr\u00f6dinger cats), together with multi-mode extensions of NOON states. We write level, one- and two-quDit reduced density matrices of symmetric <jats:italic>N<\/jats:italic>-quDit states, expressed in the last two cases in terms of collective U(<jats:italic>D<\/jats:italic>)-spin operator expectation values. Then, we evaluate level and particle entanglement for symmetric multi-quDit states with linear and von Neumann entropies of the corresponding reduced density matrices. In particular, we analyze the numerical and variational ground state of Lipkin\u2013Meshkov\u2013Glick models of 3-level identical atoms. We also propose an extension of the concept of SU(2)-spin squeezing to SU(<jats:italic>D<\/jats:italic>) and relate it to pairwise <jats:italic>D<\/jats:italic>-level atom entanglement. Squeezing parameters and entanglement entropies are good markers that characterize the different quantum phases, and their corresponding critical points, that take place in these interacting <jats:italic>D<\/jats:italic>-level atom models.<\/jats:p>","DOI":"10.1007\/s11128-021-03218-6","type":"journal-article","created":{"date-parts":[[2021,9,13]],"date-time":"2021-09-13T07:04:31Z","timestamp":1631516671000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Entanglement and U(D)-spin squeezing in symmetric multi-quDit systems and applications to quantum phase transitions in Lipkin\u2013Meshkov\u2013Glick D-level atom models"],"prefix":"10.1007","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2566-9590","authenticated-orcid":false,"given":"Manuel","family":"Calixto","sequence":"first","affiliation":[]},{"given":"Alberto","family":"Mayorgas","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7118-4500","authenticated-orcid":false,"given":"Julio","family":"Guerrero","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,9,13]]},"reference":[{"key":"3218_CR1","doi-asserted-by":"publisher","first-page":"517","DOI":"10.1103\/RevModPhys.80.517","volume":"80","author":"L Amico","year":"2008","unstructured":"Amico, L., Fazio, R., Osterloh, A., Vedral, V.: Entanglement in many-body systems. 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