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We study linear constraint system (LCS) games over the ring of arithmeticmodulo $d$. We give a new proof that certain LCS games (the Mermin--Peres MagicSquare and Magic Pentagram over binary alphabets, together with parallelrepetitions of these) have unique winning strategies, where the uniqueness isrobust to small perturbations. In order to prove our result, we extend therepresentation-theoretic framework of Cleve, Liu, and Slofstra (Journal ofMathematical Physics 58.1 (2017): 012202.) to apply to linear constraint gamesover $\mathbb{Z}_d$ for $d\geq 2$. We package our main argument into machinerywhich applies to any nonabelian finite group with a ''solution group''presentation. We equip the $n$-qubit Pauli group for $n\geq 2$ with such apresentation; our machinery produces the Magic Square and Pentagram games fromthe presentation and provides robust self-testing bounds. The question ofwhether there exist LCS games self-testing maximally entangled states of localdimension not a power of $2$ is left open. A previous version of this paperfalsely claimed to show self-testing results for a certain generalization ofthe Magic Square and Pentagram mod $d\neq 2$. We show instead that such aresult is impossible.

Robust self-testing for linear constraint system games