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The aim of the Higher-Order Weak Lensing Statistics (HOWLS) project is to assess, compare, and combine the constraining power of ten different HOS on a common set of<jats:italic>Euclid<\/jats:italic>-like mocks, derived from<jats:italic>N<\/jats:italic>-body simulations. In this first paper of the HOWLS series, we computed the nontomographic (\u03a9<jats:sub>m<\/jats:sub>,<jats:italic>\u03c3<\/jats:italic><jats:sub>8<\/jats:sub>) Fisher information for the one-point probability distribution function, peak counts, Minkowski functionals, Betti numbers, persistent homology Betti numbers and heatmap, and scattering transform coefficients, and we compare them to the shear and convergence two-point correlation functions in the absence of any systematic bias. We also include forecasts for three implementations of higher-order moments, but these cannot be robustly interpreted as the Gaussian likelihood assumption breaks down for these statistics. Taken individually, we find that each HOS outperforms the two-point statistics by a factor of around two in the precision of the forecasts with some variations across statistics and cosmological parameters. When combining all the HOS, this increases to a 4.5 times improvement, highlighting the immense potential of HOS for cosmic shear cosmological analyses with<jats:italic>Euclid<\/jats:italic>. The data used in this analysis are publicly released with the paper.<\/jats:p>","DOI":"10.1051\/0004-6361\/202346017","type":"journal-article","created":{"date-parts":[[2023,4,21]],"date-time":"2023-04-21T18:47:37Z","timestamp":1682102857000},"page":"A120","source":"Crossref","is-referenced-by-count":55,"title":["<i>Euclid<\/i>preparation"],"prefix":"10.1051","volume":"675","author":[{"name":"Euclid Collaboration","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9442-2527","authenticated-orcid":false,"given":"V.","family":"Ajani","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4145-1943","authenticated-orcid":false,"given":"M.","family":"Baldi","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1060-3959","authenticated-orcid":false,"given":"A.","family":"Barthelemy","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7537-6921","authenticated-orcid":false,"given":"A.","family":"Boyle","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8637-6305","authenticated-orcid":false,"given":"P.","family":"Burger","sequence":"additional","affiliation":[]},{"given":"V. 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