Image compression and image encryption are two essential tasks in image processing. The former aims to reduce the cost for storage or transmission of images while the latter aims to change the positions or values of pixels to protect image content. Nowadays, an increasing number of researchers are focusing on the combination of these two tasks. In this paper, we propose a novel joint image compression and encryption approach that integrates a quantum chaotic system, sparse Bayesian learning (SBL) and a bit-level 3D Arnold cat map, so-called QSBLA, for such a purpose. Specifically, the QSBLA consists of 6 stages. First, a quantum chaotic system is employed to generate chaotic sequences for subsequent compression and encryption. Second, as one method of compressive sensing, SBL is used to compress images. Third, an operation of diffusion is performed on the compressed image. Fourth, the compressed and diffused image is transformed into several bit-level cubes. Fifth, 3D Arnold cat maps are used to permute each bit-level cube. Finally, all the bit-level cubes are integrated and transformed into a 2D pixel-level image, resulting in the compressed and encrypted image. Extensive experiments on 8 publicly-accessed images demonstrate that the proposed QSBLA is superior or comparable to some state-of-the-art approaches in terms of several measurement indices, indicating that the QSBLA is promising for joint image compression and encryption.