Quantum steering is a pivotal phenomenon in quantum mechanics, allowing one party to nonlocally affect the state of another distant party's quantum system. Certifying high-dimensional quantum steering is crucial for advancing quantum communication and computation technologies. Recently, we developed semidefinite programming hierarchies for characterising high-dimensional entanglement in steering experiments. Our algorithm provide a general method for detecting genuinely high-dimensional steering, via certification of the Schmidt number of the state. The method’s key advantage is dimension-scalability: no additional computational cost is associated with testing larger Schmidt number. Secondly, we showcase how the same algorithm can be adapted to quantify the fidelity of entanglement with maximally entangled states. Practical utility is demonstrated through relevant case studies, highlighting the method applicability to state of the art experiments. The certification of high-dimensional quantum steering through SDP relaxation is computationally intensive due to several reasons. Solving SDPs involves manipulating large matrices and performing extensive linear algebra operations, which require significant computational power. Additionally, SDP problems are typically solved using iterative algorithms that converge to an optimal solution. These algorithms benefit greatly from parallel processing to expedite convergence. As the dimension of the quantum system increases, the size of the SDP grows exponentially, demanding more memory and processing power. Notably, the algorithm we developed, have already been tested on standard computers for smaller dimensions, demonstrating their correctness and efficiency. However, to address higher dimensions and provide a useful tool for state-of-the-art experiments, the computational resources of a cluster are indispensable.