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In practice, the characterization of the set <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>Q<\/mml:mi><\/mml:math> of quantum correlations is carried out, often, through a converging hierarchy of outer approximations. On the other hand, some subsets of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>Q<\/mml:mi><\/mml:math> arising from additional constraints [e.g., originating from quantum states having positive-partial-transposition (PPT) or being finite-dimensional maximally entangled (MES)] turn out to be also amenable to similar numerical characterizations. How, then, at a quantitative level, are all these naturally restricted subsets of nonsignaling correlations different? Here, we consider several bipartite Bell scenarios and numerically estimate their volume relative to that of the set of nonsignaling correlations. Within the number of cases investigated, we have observed that (1) for a given number of inputs <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mi>n<\/mml:mi><mml:mi>s<\/mml:mi><\/mml:msub><\/mml:math> (outputs <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mi>n<\/mml:mi><mml:mi>o<\/mml:mi><\/mml:msub><\/mml:math>), the relative volume of both the Bell-local set and the quantum set increases (decreases) rapidly with increasing <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mi>n<\/mml:mi><mml:mi>o<\/mml:mi><\/mml:msub><\/mml:math> (<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mi>n<\/mml:mi><mml:mi>s<\/mml:mi><\/mml:msub><\/mml:math>) (2) although the so-called macroscopically local set <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mi>Q<\/mml:mi><mml:mn>1<\/mml:mn><\/mml:msub><\/mml:math> may approximate <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>Q<\/mml:mi><\/mml:math> well in the two-input scenarios, it can be a very poor approximation of the quantum set when <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mi>n<\/mml:mi><mml:mi>s<\/mml:mi><\/mml:msub><\/mml:math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mo>&amp;#x003E;<\/mml:mo><\/mml:math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mi>n<\/mml:mi><mml:mi>o<\/mml:mi><\/mml:msub><\/mml:math> (3) the almost-quantum set <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>Q<\/mml:mi><mml:mo stretchy=\"false\">&amp;#x007E;<\/mml:mo><\/mml:mover><\/mml:mrow><mml:mn>1<\/mml:mn><\/mml:msub><\/mml:math> is an exceptionally-good approximation to the quantum set (4) the difference between <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>Q<\/mml:mi><\/mml:math> and the set of correlations originating from MES is most significant when <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mi>n<\/mml:mi><mml:mi>o<\/mml:mi><\/mml:msub><mml:mo>=<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:math>, whereas (5) the difference between the Bell-local set and the PPT set generally becomes more significant with increasing <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mi>n<\/mml:mi><mml:mi>o<\/mml:mi><\/mml:msub><\/mml:math>. This last comparison, in particular, allows us to identify Bell scenarios where there is little hope of realizing the Bell violation by PPT states and those that deserve further exploration.<\/jats:p>","DOI":"10.22331\/q-2022-07-14-765","type":"journal-article","created":{"date-parts":[[2022,7,14]],"date-time":"2022-07-14T13:28:36Z","timestamp":1657805316000},"page":"765","update-policy":"http:\/\/dx.doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":8,"title":["Naturally restricted subsets of nonsignaling correlations: typicality and convergence"],"prefix":"10.22331","volume":"6","author":[{"given":"Pei-Sheng","family":"Lin","sequence":"first","affiliation":[{"name":"Department of Physics and Center for Quantum Frontiers of Research & Technology (QFort), National Cheng Kung University, Tainan 701, Taiwan"}]},{"given":"Tam\u00e1s","family":"V\u00e9rtesi","sequence":"additional","affiliation":[{"name":"MTA Atomki Lend\u00fclet Quantum Correlations Research Group, Institute for Nuclear Research, P.O. Box 51, H-4001 Debrecen, Hungary"}]},{"given":"Yeong-Cherng","family":"Liang","sequence":"additional","affiliation":[{"name":"Department of Physics and Center for Quantum Frontiers of Research & Technology (QFort), National Cheng Kung University, Tainan 701, Taiwan"},{"name":"Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan"}]}],"member":"9598","published-online":{"date-parts":[[2022,7,14]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"A. Ac\u00edn. Statistical distinguishability between unitary operations. Phys. Rev. Lett., 87: 177901, Oct 2001. 10.1103\/PhysRevLett.87.177901.","DOI":"10.1103\/PhysRevLett.87.177901"},{"key":"1","unstructured":"Antonio Ac\u00edn. (private communication)."},{"key":"2","doi-asserted-by":"publisher","unstructured":"Antonio Ac\u00edn, Nicolas Brunner, Nicolas Gisin, Serge Massar, Stefano Pironio, and Valerio Scarani. Device-independent security of quantum cryptography against collective attacks. 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