Contextuality: a benchmark for ``quantumness'' in theory and experiment

An ontological model for a quantum experiment is a purported explanation for the probabilities we see. Non-contextual models are particularly compelling because they never offer two different explanations for the same observations. I will review these notions and argue that the impossibility of non-contextual models is a good way to rigorously define ``genuine quantumness''. Two examples of phenomena whose ``quantumness'' have been debated are ``anomalous weak values'' and ``logical pre- and post-selection paradoxes''. I will outline how their incompatibility with non-contextual models clarifies why both defy compelling classical explanation. A related idea is to use contextuality to certify that an experiment has achieved quantum coherence without requiring a full characterization. To this end I will present a simple yet robust non-contextuality inequality that can be violated with a single qubit. (The work on pre- and post-selection paradoxes was done in collaboration with with Matt Leifer.)