  Quantum superposition states are behind many of the curious phenomena
exhibited by quantum systems, including Bell non-locality, quantum
interference, quantum computational speed-up, and the measurement problem. At
the same time, many qualitative properties of quantum superpositions can also
be observed in classical probability distributions leading to a suspicion that
superpositions may be explicable as probability distributions over less
problematic states, that is, a suspicion that superpositions are
\emph{epistemic}. Here, it is proved that, for any quantum system of dimension
$d>3$, this cannot be the case for almost all superpositions. Equivalently, any
underlying ontology must contain ontic superposition states. A related question
concerns the more general possibility that some pairs of non-orthogonal quantum
states $|\psi\rangle,|\phi\rangle$ could be ontologically indistinct (there are
ontological states which fail to distinguish between these quantum states). A
similar method proves that if $|\langle\phi|\psi\rangle|^{2}\in(0,\frac{1}{4})$
then $|\psi\rangle,|\phi\rangle$ must approach ontological distinctness as
$d\rightarrow\infty$. The robustness of these results to small experimental
error is also discussed.
