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A hierarchy of semidefinite programming (SDP) relaxations approximates the global optimum of polynomial optimization problems of noncommuting variables. Typical applications include determining genuine multipartite entanglement, finding mutually unbiased bases, and solving ground-state energy problems. Generating the relaxations, however, is a computationally demanding task, and only problems of commuting variables have efficient generators. A closer look at the applications reveals sparse structures: either the monomial base can be reduced or the moment matrix has an inherent structure. Exploiting sparsity leads to scalable generation of relaxations, and the resulting SDP problem is also easier to solve. The current algorithm is able to generate relaxations of optimization problems of a hundred noncommuting variables and a quadratic number of constraints.