  We present a technique for reducing the computational requirements by several
orders of magnitude in the evaluation of semidefinite relaxations for bounding
the set of quantum correlations arising from finite-dimensional Hilbert spaces.
The technique, which we make publicly available through a user-friendly
software package, relies on the exploitation of symmetries present in the
optimisation problem to reduce the number of variables and the block sizes in
semidefinite relaxations. It is widely applicable in problems encountered in
quantum information theory and enables computations that were previously too
demanding. We demonstrate its advantages and general applicability in several
physical problems. In particular, we use it to robustly certify the
non-projectiveness of high-dimensional measurements in a black-box scenario
based on self-tests of $d$-dimensional symmetric informationally complete
POVMs.
