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Traditional concentrating photovoltaic (CPV) systems utilize multijunction cells to minimize thermalization losses, but cannot efficiently capture diffuse sunlight, which contributes to a high levelized cost of energy (LCOE) and limits their use to geographical regions with high direct sunlight insolation. Luminescent solar concentrators (LSCs) harness light generated by luminophores embedded in a light-trapping waveguide to concentrate light 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 PLMA polymer layer with embedded CdSe/CdS quantum dot (QD) luminophores and InGaP micro-cells, which serve as a high bandgap absorber on top of a conventional Si photovoltaic. We experimentally synthesize CdSe/CdS QDs with exceptionally high quantum-yield (99%) and ultra-narrowband emission optimally matched to fabricated III-V InGaP micro-cells. Using a Monte Carlo ray-tracing model, we show the radiative limit power conversion efficiency for a module with these components to be 30.8% diffuse sunlight conditions. These results indicate that a tandem LSC-on-Si architecture could significantly improve upon the efficiency of a conventional Si photovoltaic module with simple and straightforward alterations of the module lamination steps of a Si photovoltaic manufacturing process, with promise for widespread module deployment across diverse geographical regions and energy markets.