Quantum correlations cannot be reproduced with a finite number of measurements in any no-signaling theory

We show, for any finite $n \geq 2$, that there exist quantum correlations obtained from performing $n$ dichotomic quantum measurements in a bipartite Bell scenario, which cannot be reproduced by mixtures of measurement devices with at most $(n-1)$ incompatible measurements across different partitions in any no-signaling theory. That is, it requires any no-signaling theory an unbounded number of measurements to reproduce the predictions of quantum theory. We prove our results by showing that there exist linear Bell inequalities that have to be obeyed by any no-signaling theory involving only $(n-1)$-wise incompatible measurements and show explicitly how these can be violated in quantum theory. Finally, we discuss the relation of our work to previous works ruling out alternatives to quantum theory with some kind of bounded degree of freedom and consider the experimental verifiability of our results.