  Spekkens' toy theory is a non-contextual hidden variable model with an
epistemic restriction, a constraint on what the observer can know about the
reality. It has been shown in [3] that for qudits of odd dimensions it is
operationally equivalent to stabiliser quantum mechanics by making use of
Gross' theory of discrete Wigner functions. This result does not hold in the
case of qubits, because of the unavoidable negativity of any Wigner function
representation of qubit stabiliser quantum mechanics. In this work we define
and characterise the subtheories of Spekkens' theory that are operationally
equivalent to subtheories of stabiliser quantum mechanics. We use these
Spekkens' subtheories as a unifying framework for the known examples of
state-injection schemes where contextuality is an injected resource to reach
universal quantum computation. In addition, we prove that, in the case of
qubits, stabiliser quantum mechanics can be reduced to a Spekkens' subtheory in
the sense that all its objects that do not belong to the Spekkens' subtheory,
namely non-covariant Clifford gates, can be injected. This shows that within
Spekkens' subtheories we possess the toolbox to perform state-injection of
every object outside of them and it suggests that there is no need to use
bigger subtheories to reach universal quantum computation via state-injection.
We conclude with a novel scheme of computation suggested by our approach which
is based on the injection of CCZ states and we also relate different proofs of
contextuality to different state injections of non-covariant gates.
