ABSTRACT. The precise features of quantum theory enabling quantum computational advantage are unclear. Contextuality has emerged as a promising hypothesis: e.g. the magic states needed to practically achieve quantum computation are contextual.

Strong contextuality, as defined by Abramsky-Brandenburger, is an extremal form of contextuality describing systems that exhibit logically paradoxical behaviour.

After introducing number-theoretic techniques for constructing exotic quantum paradoxes, we give large families of strongly contextual magic states that are optimal in the sense that they enable deterministic injection of gates of the Clifford hierarchy. We thereby bolster a refinement of the resource theory of contextuality that emphasises the computational power of logical paradoxes.