Low energy experiments that search for Beyond the Standard Model (BSM) physics often rely on nuclear targets. Therefore, it is imperative that we obtain a clear theoretical picture of the nuclear physics involved. Effective field theory provides a model-independent framework to capture the nuclear physics in terms of few-nucleon currents. However, every operator in an effective theory is accompanied by an undetermined low energy coefficient that must be determined from data or a nonperturbative quantum chromodynamics calculation such as a lattice calculation. For many processes, these determinations are not yet possible; thus, other theoretical constraints are necessary in order to guide the interpretation of experimental bounds. Here, we review recent constraints obtained from the large-$N_c$ limit of QCD, where $N_c$ is the number of colors, for BSM few-nucleon currents relevant for neutrinoless double beta decay and dark matter direct detection.