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The interplay between the quantum state space and a specific set ofmeasurements can be effectively captured by examining the set of jointlyattainable expectation values. This set is commonly referred to as the (convex)joint numerical range. In this work, we explore geometric properties of thisconstruct for measurements represented by tensor products of Pauli observables,also known as Pauli strings. The structure of pairwise commutation andanticommutation relations among a set of Pauli strings determines a graph $G$,sometimes also called the frustration graph. We investigate the connectionbetween the parameters of this graph and the structure of minimal ellipsoidsencompassing the joint numerical range. Such an outer approximation can be verypractical since ellipsoids can be handled analytically even in high dimensions.  We find counterexamples to a conjecture from [C. de Gois, K. Hansenne and O.G\"uhne, arXiv:2207.02197], and answer an open question in [M. B. Hastings andR. O'Donnell, Proc. STOC 2022, pp. 776-789], which implies a new graphparameter that we call $\beta(G)$. Besides, we develop this approach indifferent directions, such as comparison with graph-theoretic approaches inother fields, applications in quantum information theory, numerical methods,properties of the new graph parameter, etc. Our approach suggests many openquestions that we discuss briefly at the end.

Bounding the joint numerical range of Pauli strings by graph parameters