Semi-device-independent randomness generation protocols based on Kochen-Specker contextuality offer the attractive features of compact devices, high rates, and ease of experimental implementation over fully device-independent (DI) protocols. Here, we investigate this paradigm and derive four results to improve the state of the art. Firstly, we introduce a family of simple, experimentally feasible orthogonality graphs (measurement compatibility structures) for which the maximum violation of the corresponding noncontextuality inequalities allows the certification of the maximum amount of ${\log }_{2}d$ bits of randomness from a qudit system with projective measurements for $d\ge 3$. We analytically derive the Lovász theta and fractional packing number for this graph family, and thereby prove their utility for optimal randomness generation in both randomness expansion and amplification tasks. Secondly, a central additional assumption in contextuality-based protocols over fully DI ones is that the measurements are repeatable and satisfy an intended compatibility structure. We frame a relaxation of this condition in terms of $ϵ$-orthogonality graphs for a parameter $ϵ>0$, and derive quantum correlations that allow the certification of randomness for arbitrary relaxation $ϵ\in [0,1)$. Thirdly, it is well-known that a single qubit is noncontextual, i.e., the qubit correlations can be explained by a noncontextual hidden variable model. We show however that a single qubit is almost contextual in that there exist qubit correlations that cannot be explained by $ϵ$-faithful noncontextual hidden variable models for small $ϵ>0$. Finally, we point out possible attacks by quantum and general consistent (nonsignaling) adversaries for certain classes of contextuality tests over and above those considered in DI scenarios.

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• Quantum communication, protocols & technology
• Quantum correlations in quantum information
• Quantum correlations, foundations & formalism
• Quantum information processing
• Quantum information theory