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Bell non-locality and Kochen-Specker (KS) contextuality are logicallyindependent concepts, fuel different protocols with quantum vs classicaladvantage, and have distinct classical simulation costs. A natural question iswhat are the relations between these concepts, advantages, and costs. Toaddress this question, it is useful to have a map that captures all theconnections between Bell non-locality and KS contextuality in quantum theory.The aim of this work is to introduce such a map. After defining thetheory-independent notions of Bell non-locality and KS contextuality for idealmeasurements, we show that, in quantum theory, due to Neumark's dilationtheorem, every matrix of quantum Bell non-local correlations can be mapped toan identical matrix of KS contextual correlations produced in a scenario withidentical relations of compatibility but where measurements are ideal and nospace-like separation is required. A more difficult problem is identifyingconnections in the opposite direction. We show that there are "one-to-one" andpartial connections between KS contextual correlations and Bell non-localcorrelations for some KS contextuality scenarios, but not for all of them.However, there is also a method that transforms any matrix of KS contextualcorrelations for quantum systems of dimension $d$ into a matrix of Bellnon-local correlations between two quantum subsystems each of them of dimension$d$. We collect all these connections in map and list some problems which canbenefit from this map.

Bell non-locality and Kochen-Specker contextuality: How are they connected?