  While quantum computers are expected to yield considerable advantages over
classical devices, the precise features of quantum theory enabling these
advantages remain unclear. Contextuality--the denial of a notion of classical
physical reality--has emerged as a promising hypothesis.
  Magic states are quantum resources critical for practically achieving
universal quantum computation. They exhibit the standard form of contextuality
that is known to enable probabilistic advantages in a variety of computational
and communicational tasks. Strong contextuality is an extremal form of
contextuality describing systems that exhibit logically paradoxical behaviour.
  Here, we consider special magic states that deterministically enable quantum
computation. After introducing number-theoretic techniques for constructing
exotic quantum paradoxes, we present large classes of strongly contextual magic
states that enable deterministic implementation of gates from the Clifford
hierarchy. These surprising discoveries bolster a refinement of the resource
theory of contextuality that emphasises the computational power of logical
paradoxes.
