  For certain restricted computational tasks, quantum mechanics provides a
provable advantage over any possible classical implementation. Several of these
results have been proven using the framework of measurement-based quantum
computation (MBQC), where non-locality and more generally contextuality have
been identified as necessary resources for certain quantum computations. Here,
we consider the computational power of MBQC in more detail by refining its
resource requirements, both on the allowed operations and the number of
accessible qubits. More precisely, we identify which Boolean functions can be
computed in non-adaptive MBQC, with local operations contained within a finite
level in the Clifford hierarchy. Moreover, for non-adaptive MBQC restricted to
certain subtheories such as stabiliser MBQC, we compute the minimal number of
qubits required to compute a given Boolean function. Our results point towards
hierarchies of resources that more sharply characterise the power of MBQC
beyond the binary of contextuality vs non-contextuality.
