The maximum stable set problem is a well-known NP-hard problem in combinatorial optimization, which can be reformulated as the maximization of a quadratic square-free polynomial over the (Boolean) hypercube. A popular approach to polynomial optimization problems is to build a hierarchy of convex tractable relaxations, based on replacing the (hard to test) positivity condition for a polynomial by a tractable, sufficient condition for positivity. One may for instance search for positivity certificates in the form of a combination of products of constraints where the multipliers could be sums of squares of polynomials or simply nonnegative numbers. This leads respectively to SDP and LP relaxations. Although SDP hierarchies are stronger, they are more difficult to analyze and computationally more expensive. This motivates our study of the linear programming hierarchy for the maximum stable set problem, based on Handelman's representation result for positive polynomials on a polytope. Using Bernstein polynomials Park and Hong (2012) could show some error bounds for the approximate solutions obtained at any order in the hierarchy. We further investigate this hierarchy. In particular, we show a relation to fractional clique covers of graphs, we give bounds for the rank of the Handelman hierarchy (i.e., the smallest order needed to obtain the true optimum), and we point out links to some other LP and SDP hierarchies (of Sherali-Adams, Lasserre, and Lovasz-Schrijver).