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Optim."],"published-print":{"date-parts":[[2026,3,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>We revisit the convergence analysis of two approximation hierarchies for polynomial optimization on the unit sphere.\u00a0The first one is based on the moment-sos approach and gives semidefinite bounds for which Fang and Fawzi (2021) showed an analysis in [Formula: see text] for the [Formula: see text]th level bound, using the polynomial kernel method. The second hierarchy was recently proposed by Lovitz and Johnston (2023) and gives spectral bounds for which they show a convergence rate in [Formula: see text], using a quantum de Finetti theorem of Christandl et\u00a0al.\u00a0(2007) that applies to complex Hermitian matrices with a \u201cdouble\u201d symmetry. We investigate links between these approaches, in particular, via duality of moments and sums of squares. Our main results include showing that the spectral bounds cannot have a convergence rate better than [Formula: see text] and that they do not enjoy generic finite convergence. In addition, we propose alternative performance analyses that involve explicit constants depending on intrinsic parameters of the optimization problem. For this we develop a novel \u201cbanded\u201d real de Finetti theorem that applies to real matrices with \u201cdouble\u201d symmetry. We also show how to use the polynomial kernel method to obtain a de Finetti type result in [Formula: see text] for real maximally symmetric matrices, improving an earlier result in [Formula: see text] of Doherty and Wehner (2012).<\/jats:p>","DOI":"10.1137\/24m1717750","type":"journal-article","created":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T08:42:07Z","timestamp":1771576927000},"page":"204-232","source":"Crossref","is-referenced-by-count":1,"title":["Moment-Sos and Spectral Hierarchies for Polynomial Optimization on the Sphere and Quantum de Finetti Theorems"],"prefix":"10.1137","volume":"36","author":[{"given":"Alexander Taveira","family":"Blomenhofer","sequence":"first","affiliation":[{"name":"QMATH, University of Copenhagen, 2100 Copenhagen."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Monique","family":"Laurent","sequence":"additional","affiliation":[{"name":"Centrum Wiskunde & Informatica (CWI), 1090 GB Amsterdam, and Tilburg University."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2026,2,20]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-011-0499-2"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1007\/s13324-023-00844-4"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-022-01877-6"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2024.08.022"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1137\/23M1555430"},{"key":"ref6","unstructured":"A. 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