Recently we became aware of an important reference that was published during the preparations of our manuscript, which we failed to cite in the original paper. In Ref. [1], the authors propose a similar scheme for the generation of multiple entangled pairs between qubit registers using a high-dimensional photonic qudit and cavity-mediated spin-photon gates. Contrary to Ref. [1], we show that such photonic qudit-mediated entanglement generation schemes have similar distribution rates as standard (parallel) qubit approaches but the memory requirements are significantly relaxed for the qudit schemes. ... Read more