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The historical significance of the Stern-Gerlach experiment lies in its provision of the initial evidence for space quantization. Over time, its sequential form has evolved into an elegant paradigm that effectively illustrates the fundamental principles of quantum theory. To date, the practical implementation of the sequential Stern-Gerlach experiment has not been fully achieved. In this study, we demonstrate the capability of programmable quantum processors to simulate the sequential Stern-Gerlach experiment. The specific parametric shallow quantum circuits, which are suitable for the limitations of current noisy quantum hardware, are given to replicate the functionality of Stern-Gerlach devices with the ability to perform measurements in different directions. Surprisingly, it has been demonstrated that Wigner's Stern-Gerlach interferometer can be readily implemented in our sequential quantum circuit. With the utilization of the identical circuits, it is also feasible to implement Wheeler's delayed-choice experiment. We propose the utilization of cross-shaped programmable quantum processors to showcase sequential experiments, and the simulation results demonstrate a strong alignment with theoretical predictions. With the rapid advancement of cloud-based quantum computing, such as BAQIS Quafu, it is our belief that the proposed solution is well-suited for deployment on the cloud, allowing for public accessibility. Our findings not only expand the potential applications of quantum computers, but also contribute to a deeper comprehension of the fundamental principles underlying quantum theory.