We describe the design of a cryogenic rotation stage (CRS) for use with the cryogenic half-wave plate (CHWP) polarization modulator on the POLARBEAR-2b and POLARBEAR-2c (PB2b/c) cosmic microwave background (CMB) experiments, the second and third installments of the Simons Array. Rapid modulation of the CMB polarization signal using a CHWP suppresses 1/f contamination due to atmospheric turbulence and allows a single polarimeter to measure both polarization states, mitigating systematic effects that arise when differencing orthogonal detectors. To modulate the full detector array while avoiding excess photon loading due to thermal emission, the CHWP must have a clear-aperture diameter of > 450 mm and be cooled to < 100 K. We have designed a 454-mm-clear-aperture, < 65 K CRS using a superconducting magnetic bearing driven by a synchronous magnetic motor. We present the specifications for the CRS, its interfacing to the PB2b/c receiver cryostat, its performance in a stand-alone test, and plans for future work.Cosmic microwave background (CMB) polarization is a powerful probe of cosmology. Particularly, the "B-mode" CMB polarization pattern can be used to probe gravitational lensing on arcminute angular scales [1, 2] and primordial gravitational waves on degree angular scales [3][4][5]. In order for a single telescope to characterize small and large scales simultaneously, it must observe with high resolution over a large sky area, requiring both a large primary aperture and good low-frequency (or "1/f") noise performance [6].Rapidly-rotating half-wave plates (HWP) are a common technique to modulate CMB polarization and reduce the impact of low-frequency noise on experiment sensitivity [6][7][8][9][10][11][12].