Hierarchies of resources for measurement-based quantum computation

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 measurement-based computation (MBQC), where non-locality and more generally contextuality identified as necessary resources for computations. Here, we consider power MBQC in detail by refining its resource requirements, both on allowed operations number accessible qubits. More precisely, identify which Boolean functions can be computed non-adaptive MBQC, with local contained within finite level Clifford hierarchy. Moreover, to subtheories such stabiliser compute minimal qubits required given function. Our point towards hierarchies that sharply characterise beyond binary vs non-contextuality.