The ability to control multidimensional quantum systems is key for the investigation of fundamental science and for the development of advanced quantum technologies. Here we demonstrate a multidimensional integrated quantum photonic platform able to robustly generate, control and analyze high-dimensional entanglement. We realize a programmable bipartite entangled system with dimension up to $15 \times 15$ on a large-scale silicon-photonics quantum circuit. The device integrates more than 550 photonic components on a single chip, including 16 identical photon-pair sources. We verify the high precision, generality and controllability of our multidimensional technology, and further exploit these abilities to demonstrate key quantum applications experimentally unexplored before, such as quantum randomness expansion and self-testing on multidimensional states. Our work provides a prominent experimental platform for the development of multidimensional quantum technologies.