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In 1960, the mathematician Ernst Specker described a simple example ofnonclassical correlations which he dramatized using a parable about a seer whosets an impossible prediction task to his daughter's suitors. We revisit thisexample here, using it as an entree to three central concepts in quantumfoundations: contextuality, Bell-nonlocality, and complementarity.Specifically, we show that Specker's parable offers a narrative thread thatweaves together a large number of results, including: the impossibility ofmeasurement-noncontextual and outcome-deterministic ontological models ofquantum theory (the Kochen-Specker theorem), in particular the proof ofKlyachko; the impossibility of Bell-local models of quantum theory (Bell'stheorem), especially the proofs by Mermin and Hardy; the impossibility of apreparation-noncontextual ontological model of quantum theory; and theexistence of triples of positive operator valued measures (POVMs) that can bemeasured jointly pairwise but not triplewise. Along the way, several novelresults are presented, including: a generalization of a theorem by Fineconnecting the existence of a joint distribution over outcomes ofcounterfactual measurements to the existence of a noncontextual model; ageneralization of Klyachko's proof of the Kochen-Specker theorem; a proof ofthe Kochen-Specker theorem in the style of Hardy's proof of Bell's theorem; acategorization of contextual and Bell-nonlocal correlations in terms offrustrated networks; a new inequality testing preparation noncontextuality; andlastly, some results on the joint measurability of POVMs and the question ofwhether these can be modeled noncontextually. Finally, we emphasize thatSpecker's parable provides a novel type of foil to quantum theory, challengingus to explain why the particular sort of contextuality and complementarityembodied therein does not arise in a quantum world.

Specker's Parable of the Over-protective Seer: A Road to Contextuality, Nonlocality and Complementarity (PLUS AN ERRATUM)