  In 1960, the mathematician Ernst Specker described a simple example of
nonclassical correlations which he dramatized using a parable about a seer who
sets an impossible prediction task to his daughter's suitors. We revisit this
example here, using it as an entree to three central concepts in quantum
foundations: contextuality, Bell-nonlocality, and complementarity.
Specifically, we show that Specker's parable offers a narrative thread that
weaves together a large number of results, including: the impossibility of
measurement-noncontextual and outcome-deterministic ontological models of
quantum theory (the Kochen-Specker theorem), in particular the proof of
Klyachko; the impossibility of Bell-local models of quantum theory (Bell's
theorem), especially the proofs by Mermin and Hardy; the impossibility of a
preparation-noncontextual ontological model of quantum theory; and the
existence of triples of positive operator valued measures (POVMs) that can be
measured jointly pairwise but not triplewise. Along the way, several novel
results are presented, including: a generalization of a theorem by Fine
connecting the existence of a joint distribution over outcomes of
counterfactual measurements to the existence of a noncontextual model; a
generalization of Klyachko's proof of the Kochen-Specker theorem; a proof of
the Kochen-Specker theorem in the style of Hardy's proof of Bell's theorem; a
categorization of contextual and Bell-nonlocal correlations in terms of
frustrated networks; a new inequality testing preparation noncontextuality; and
lastly, some results on the joint measurability of POVMs and the question of
whether these can be modeled noncontextually. Finally, we emphasize that
Specker's parable provides a novel type of foil to quantum theory, challenging
us to explain why the particular sort of contextuality and complementarity
embodied therein does not arise in a quantum world.
