In this work, we introduce a method to express the stabilizer of a $\mathbf{k}$-uniform complete hypergraph state as a linear combination of local operators. Quantum hypergraph states generalize graph states and exhibit properties that are not shared by their graph counterparts—most notably, their stabilizers are intrinsically nonlocal, as hyperedges can involve arbitrary subsets of vertices. Our formulation provides an explicit description of the stabilizers for $\mathbf{k}$-uniform complete hypergraphs and may offer insights for exploring these states within the stabilizer formalism. In particular, this approach could facilitate the construction of new Bell inequalities or find applications in quantum error correction.

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• Quantum correlations, foundations & formalism
• Quantum gates
• Quantum information theory