The proper computation of the time evolution of the fracture front is the main challenge of three-dimensional (3D) hydraulic fracture growth simulation. We discuss explicit and implicit variants of a hydraulic fracture propagation scheme based on a level set representation of the fracture. Such a scheme couples a finite discretization of the governing equations and the near-tip hydraulic fracture asymptotes. We benchmark the accuracy, robustness, and stability of these different front advancing schemes on a number of test cases. Our results indicate a large computational gain of the explicit scheme at the expense of a slightly less accurate solution (few percent less accuracy over few time steps) when crossing heterogeneities. The predictor corrector scheme combines at least an approximately 25% computational gain while retaining the stability and accuracy of the fully implicit version of the scheme in all cases.