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Linear constraint systems (LCS) have proven to be a surprisingly prolifictool in the study of non-classical correlations and various related issues inquantum foundations. Many results are known for the Boolean case, yet thegeneralisation to systems of odd dimension is largely open. In particular, itis not known whether there exist LCS in odd dimension, which admitfinite-dimensional quantum, but no classical solutions.  Here, we approach this question from a computational perspective. We observethat every deterministic, non-adaptive measurement-based quantum computation(MBQC) with linear side-processing defines a LCS. Moreover, the measurementoperators of such a MBQC almost define a quantum solution to the respectiveLCS: the only difference is that measurement operators generally only commutewith respect to the resource state of the MBQC. This raises the questionwhether this state-dependence can be lifted in certain cases, thus providingexamples of quantum solutions to LCS in odd dimension. Our main result assertsthat no such examples arise within a large extension of the Pauli group for podd prime, which naturally arises from and is universal for computation indeterministic, non-adaptive MBQC with linear side-processing.

No state-independent contextuality can be extracted from contextual measurement-based quantum computation with qudits of odd prime dimension