Luminescent solar concentrators (LSCs) harness light generated by luminophores embedded in a light-trapping waveguide to concentrate onto smaller cells. LSCs can absorb both direct and diffuse sunlight, and thus can operate as flat plate receivers at a fixed tilt and with a conventional module form factor. However, current LSCs experience significant power loss through parasitic luminophore absorption and incomplete light trapping by the optical waveguide. Here, we introduce a tandem LSC device architecture that overcomes both of these limitations, consisting of a poly(lauryl methacrylate) polymer layer with embedded cadmium selenide core, cadmium sulfide shell (CdSe/CdS) quantum dot (QD) luminophores and an InGaP microcell array, which serves as high bandgap absorbers on the top of a conventional Si photovoltaic. We investigate the design space for a tandem LSC, using experimentally measured performance parameters for key components, including the InGaP microcell array, CdSe/CdS QDs, and spectrally selective waveguide filters. Using a Monte Carlo ray-tracing model, we compute the power conversion efficiency for a tandem LSC module with these components to be 29.4% under partially diffuse illumination conditions. These results indicate that a tandem LSC-on-Si architecture could significantly improve upon the efficiency of a conventional Si photovoltaic cell.