  Contextuality has been conjectured to be a super-classical resource for
quantum computation, analogous to the role of non-locality as a super-classical
resource for communication. We show that the presence of contextuality places a
lower bound on the amount of classical memory required to simulate any quantum
sub-theory, thereby establishing a quantitative connection between
contextuality and classical simulability. We apply our result to the qubit
stabilizer sub-theory, where the presence of state-independent contextuality
has been an obstacle in establishing contextuality as a quantum computational
resource. We find that the presence of contextuality in this sub-theory demands
that the minimum number of classical bits of memory required to simulate a
multi-qubit system must scale quadratically in the number of qubits; notably,
this is the same scaling as the Gottesman-Knill algorithm. We contrast this
result with the (non-contextual) qudit case, where linear scaling is possible.
