GaAsP / Si tandem cells represent a potential path to simultaneously achieve high efficiency and the requisite 25-year lifetime, both of which are necessary to attain low cost. However, threading dislocation densities (TDD) exceeding 108 cm-2 have historically limited the GaAsP top cell efficiency to <10%, which is too low to realize a tandem efficiency greater than that of Si alone. In this talk, I will discuss how we have lowered the TDD of GaAsP top cells on GaP/Si by controlling dislocation dynamics throughout the growth process to consistently achieve TDD values of 4-5×106 cm-2, slightly higher than in III-V metamorphic solar cells grown on GaAs. The lower TDD has enabled Voc values of 1.16V for 1.68 eV GaAsP top cells grown on Si, while a well-designed anti-reflection coating has enabled an NREL-certified single-junction efficiency of 15.33%. Analysis of the spectral efficiency of these GaAsP top cells unambiguously shows that their performance is strong enough to allow tandem cells with higher efficiency than Si alone; the GaAsP top cells convert visible photons more efficiently than the best Si solar cells, a necessary precondition for a high-efficiency tandem. We have also demonstrated a GaAsP-filtered Si bottom cell that has undergone III-V growth with efficiency of ~5%. I will describe the materials and device design improvements that will be necessary to obtain efficiencies of 30% and provide an update on progress towards these goals.