As with classical computers, quantum computers require error-correction schemes to reliably perform useful large-scale calculations. The nature and frequency of errors depends on the quantum computing platform, and although there is a large literature on qubit-based coding, these are often not directly applicable to devices that store information in bosonic systems such as photonic resonators. Here, we introduce a framework for constructing quantum codes defined on spheres by recasting such codes as quantum analogues of the classical spherical codes. We apply this framework to bosonic coding, and we obtain multimode extensions of the cat codes that can outperform previous constructions but require a similar type of overhead. Our polytope-based cat codes consist of sets of points with large separation that, at the same time, form averaging sets known as spherical designs. We also recast concatenations of Calderbank–Shor–Steane codes with cat codes as quantum spherical codes, which establishes a method to autonomously protect against dephasing noise.