Contextuality-the obstruction to describing quantum mechanics in a classical statistical way-has been proposed as a resource that powers quantum computing. The measurement-based model provides a concrete manifestation of contextuality as a computational resource, as follows. If local measurements on a multi-qubit state can be used to evaluate nonlinear boolean functions with only linear control processing, then this computation constitutes a proof of strong contextuality-the possible local measurement outcomes cannot all be pre-assigned. However, this connection is restricted to the special case when the local measured systems are qubits, which have unusual properties from the perspective of contextuality. A single qubit cannot allow for a proof of contextuality, unlike higher-dimensional systems, and multiple qubits can allow for stateindependent contextuality with only Pauli observables, again unlike higher-dimensional generalisations. Here we identify precisely that strong non-locality is necessary in a qudit measurement-based computation (MBC) that evaluates high-degree polynomial functions with only linear control. We introduce the concept of local universality, which places a bound on the space of output functions accessible under the constraint of single-qudit measurements. Thus, the partition of a physical system into subsystems plays a crucial role for the increase in computational power. A prominent feature of our setting is that the enabling resources for qubit and qudit MBC are of the same underlying nature, avoiding the pathologies associated with qubit contextuality.

Contextuality is a key feature of quantum mechanics, as was first brought to light by Bohr [ Albert Einstein: Philosopher-Scientist, Library of Living Philosophers Vol. VII, edited by P. A. Schilpp (Open Court, 1998), pp. 199–241] and later realized more technically by Kochen and Specker [J. Math. Mech. 17, 59 (1967)]. Isham and Butterfield put contextuality at the heart of their topos-based formalism and gave a reformulation of the Kochen–Specker theorem in the language of presheaves in Isham and Butterfield [Int. J. Theor. Phys. 37, 2669 (1998)]. Here, we broaden this perspective considerably (partly drawing on existing, but scattered results) and show that apart from the Kochen–Specker theorem, Wigner’s theorem, Gleason’s theorem, and Bell’s theorem also relate fundamentally to contextuality. We provide reformulations of the theorems using the language of presheaves over contexts and give general versions valid for von Neumann algebras. This shows that a very substantial part of the structure of quantum theory is encoded by contextuality.

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.

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