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This is done through the derivation of quantum Bell inequalities, which are related to Tsirelson's problem and have significant applications in device-independent (DI) information processing. However, determining quantum Bell inequalities is a notoriously difficult task and only isolated examples are known. In this paper, we present families of (almost-)quantum Bell inequalities and highlight four foundational and DI applications. Firstly, it is known that quantum correlations on the non-signaling boundary are of crucial importance in the task of DI randomness extraction from weak sources. In the practical Bell scenario of two players with two <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>k<\/mml:mi><\/mml:math>-outcome measurements, we derive quantum Bell inequalities that demonstrate a separation between the quantum boundary and certain portions of the boundaries of the no-signaling polytope of dimension up to <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mn>4<\/mml:mn><mml:mi>k<\/mml:mi><mml:mo>&amp;#x2212;<\/mml:mo><mml:mn>8<\/mml:mn><\/mml:math>, extending previous results from nonlocality distillation and the collapse of communication complexity. Secondly as an immediate by-product, we give a general proof of Aumann\u2019s Agreement theorem for quantum systems as well as the almost-quantum correlations, which implies Aumann\u2019s agreement theorem is a reasonable physical principle in the context of epistemics to pick out both quantum theory and almost-quantum correlations from general no-signaling theories. Thirdly, we present a family of quantum Bell inequalities in the two players with <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>m<\/mml:mi><\/mml:math> binary measurements scenarios, that we prove serve to self-test the two-qubit singlet and the corresponding <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mn>2<\/mml:mn><mml:mi>m<\/mml:mi><\/mml:math> measurements. Interestingly, this claim generalizes the result for <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>m<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:math> discovered by Tsirelson-Landau-Masanes and shows an improvement over the state-of-the-art Device-Independent Randomness-Amplification (DIRA). Lastly, we use our quantum Bell inequalities to derive the general form of the principle of no advantage in nonlocal computation, which is an information-theoretic principle that serves to characterize the quantum correlation set.<\/jats:p>","DOI":"10.22331\/q-2024-10-02-1489","type":"journal-article","created":{"date-parts":[[2024,10,2]],"date-time":"2024-10-02T12:58:16Z","timestamp":1727873896000},"page":"1489","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":["(Almost-)Quantum Bell Inequalities and Device-Independent Applications"],"prefix":"10.22331","volume":"8","author":[{"given":"Yuan","family":"Liu","sequence":"first","affiliation":[{"name":"School of Computing and Data Science, The University of Hong Kong, Pokfulam Road, Hong Kong"}]},{"given":"Ho Yiu","family":"Chung","sequence":"additional","affiliation":[{"name":"School of Computing and Data Science, The University of Hong Kong, Pokfulam Road, Hong Kong"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1119-8721","authenticated-orcid":false,"given":"Ravishankar","family":"Ramanathan","sequence":"additional","affiliation":[{"name":"School of Computing and Data Science, The University of Hong Kong, Pokfulam Road, Hong Kong"}]}],"member":"9598","published-online":{"date-parts":[[2024,10,2]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"J. 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