Cite this problem as Problem 34.
Given a nonlocality scenario, with parties, settings and outcomes, consider the set of probability distributions of the form
This is the multipartite analog of the set of quantum correlations defined in Problem 33. The question is whether the closure of in each nonlocality scenario is a semi-algebraic set, or, in other words, whether, for any , there is a finite set of polynomials such that iff for .
For , the answer is positive. Indeed, as shown by Tsirelson  (for ) and independently by Masanes  (for all ), in this Bell scenario the extreme points of the quantum set can be realized by conducting projective measurements over an -qubit pure state. Since the dimension of the real space where lives is , by Caratheodory’s theorem , each point in must be a convex combination of at most of these extreme points.
Hence in this scenario the set can be seen as the projection of a large set of variables* subject to a finite number of polynomial constraints** to the space. By the Tarski-Seidenberg theorem , this implies that the probabilities themselves are characterized by a finite set of polynomial inequalities.
*Namely, , the matrix entries of the projectors and the state for each of the extreme points, together with their weights.
**Namely, the polynomials which constrain the matrices of each extreme point to be, respectively, projectors and normalized quantum states; the weights of the different points to be non-negative and add up to ; and the constraint that applying the Born rule over the weighted ensemble of extreme points we recover the probabilities .
 B.S. Cirel’son, Letters in Mathematical Physics 4, 93-100 (1980).
 Ll. Masanes, arXiv:quant-ph/0512100.
 Eggleston, H. G. (1958). Convexity. Cambridge University Press.
 J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 36, Springer Berlin Heidelberg, 1998.