Achieving global space-air-ground integrated network (SAGIN) access only with CubeSats presents significant challenges such as the access sustainability limitations in specific regions (e.g., polar regions) and the energy efficiency limitations in CubeSats. To tackle these problems, high-altitude long-endurance unmanned aerial vehicles (HALE-UAVs) can complement these CubeSat shortcomings for providing cooperatively global access sustainability and energy efficiency. However, as the number of CubeSats and HALE-UAVs, increases, the scheduling dimension of each ground station (GS) increases. As a result, each GS can fall into the curse of dimensionality, and this challenge becomes one major hurdle for efficient global access. Therefore, this paper provides a quantum multi-agent reinforcement Learning (QMARL)-based method for scheduling between GSs and CubeSats/HALE-UAVs in order to improve global access availability and energy efficiency. The main reason why the QMARL-based scheduler can be beneficial is that the algorithm facilitates a logarithmic-scale reduction in scheduling action dimensions, which is one critical feature as the number of CubeSats and HALE-UAVs expands. Additionally, individual GSs have different traffic demands depending on their locations and characteristics, thus it is essential to provide differentiated access services. The superiority of the proposed scheduler is validated through data-intensive experiments in realistic CubeSat/HALE-UAV settings.