The incompatibility of quantum measurements—i.e., the fact that certain observable quantities cannot be measured jointly—is widely regarded as a distinctive quantum feature with important implications for both the foundations and applications of quantum theory. While the standard notion of measurement incompatibility has been a focus of attention since the inception of quantum theory, its generalizations—such as measurement simulability, n-wise incompatibility, and multi-copy incompatibility—have only recently been proposed. In this talk, I will argue that all these generalizations reflect different ways of addressing the same question: how many distinct measurements are genuinely contained in a given measurement device? I will show that these notions differ not only in their operational meaning but also mathematically, in terms of the sets of measurement assemblages they characterize. I will then present how the relationships between these different generalizations can be fully resolved by establishing a strict hierarchy among them. Furthermore, I will discuss how to certify, in a device- and theory-independent manner, any number n >= 2 of measurements in a Bell experiment. More specifically, I will show that there exist quantum correlations obtained from performing n dichotomic quantum measurements in a bipartite Bell scenario that cannot be reproduced by (n−1) measurements in any no-signaling theory. In other words, reproducing the predictions of quantum theory requires an unbounded number of measurements in any no-signaling framework.

The talk is based on these two works:

(1) https://arxiv.org/abs/2506.21223

(2) https://journals.aps.org/prl/abstract/10.1103/7vjx-62ng