Contextuality, Complementarity, Signaling, and Bell Tests.

This is a review devoted to the complementarity-contextuality interplay with connection to the Bell inequalities. Starting the discussion with complementarity, I point to contextuality as its seed. is the dependence of an observable's outcome on the experimental context; on the system-apparatus interaction. Probabilistically, complementarity means that the (JPD) does not exist. Instead of the JPD, one has to operate with contextual probabilities. The Bell inequalities are interpreted as the statistical tests of contextuality, and hence, incompatibility. For context-dependent probabilities, these inequalities may be violated. I stress that contextuality tested by the Bell inequalities is the so-called (JMC), the special case of Bohr's contextuality. Then, I examine the role of signaling (marginal inconsistency). In QM, signaling can be considered as an experimental artifact. However, often, experimental data have signaling patterns. I discuss possible sources of signaling-for example, dependence of the state preparation on measurement settings. In principle, one can extract the measure of "pure contextuality" from data shadowed by signaling. This theory is known as (CbD). It leads to inequalities with an additional term quantifying signaling: Bell-Dzhafarov-Kujala inequalities.