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The Kochen-Specker theorem shows the impossibility for a hidden variabletheory to consistently assign values to certain (finite) sets of observables ina way that is non-contextual and consistent with quantum mechanics. If werequire non-contextuality, the consequence is that many observables must nothave pre-existing definite values. However, the Kochen-Specker theorem does notallow one to determine which observables must be value indefinite. In thispaper we present an improvement on the Kochen-Specker theorem which allows oneto actually locate observables which are provably value indefinite. Varioustechnical and subtle aspects relating to this formal proof and its connectionto quantum mechanics are discussed. This result is then utilized for theproposal and certification of a dichotomic quantum random number generatoroperating in a three-dimensional Hilbert space.

Strong Kochen-Specker theorem and incomputability of quantum randomness