The Navier–Stokes–Cahn–Hilliard (NSCH) diffuse interface model is a phase-field model that describes binary-fluid systems. This model is inherently capable of dealing with topological changes due to its phase-field nature and therefore particularly suitable for complex multi-phase flow. Recently, an advancement has been made to include complex domains using an immersed method (Stoter, Sluijs, Brummelen, Verhoosel, & Demont, 2023). We numerically approximate the NSCH model by means of an isogeometric analysis method with adaptive spatial resolution based on a residual-based error estimate. In addition, we use ε-continuation for robust spatial-refinements and to decouple the interface movement per time step from the interface width (Van Brummelen, Demont, & van Zwieten, 2020). In this model, the interface length scale is minuscule compared to the system length scale while encompassing the most complex dynamics. Therefore, this relatively small part of the domain carries a large part of the computational cost. In two-step time discretization methods, spatial-refinement is needed in the area where the interface is located at the beginning and at the end of the time step. We call this the area of common refinement. In a multi-step time discretization method, e.g. Runge-Kutta methods, the area of common refinement is much larger as it includes the interface location during the sub-increments of the time step. This results in high computational costs for these multi-step methods. Therefore, in this work, we present higher-order two-step time discretization methods for the NSCH model based on the generalized-α method as presented in (Behnoudfar, Deng, & Calo, 2020). References: Behnoudfar, P., Deng, Q., & Calo, V. (2020, 10). High-order generalized-alpha method. Applications in Engineering Science, 4 , 100021. doi: 10.1016/j.apples.2020.1000212 Stoter, S. K. F., Sluijs, T. B. V., Brummelen, E. H. V., Verhoosel, C. V., & Demont, T. H. B. (2023). Stabilized immersed isogeometric analysis for the navier-stokes-cahn-hilliard equations, with applications to binary-fluid flow through porous media. Retrieved from https://www.researchgate.net/publication/371158914 doi:10.48550/arXiv.2305.19237 Van Brummelen, E. H., Demont, T. H., & van Zwieten, G. J. (2020). An adaptive isogeometric analysis approach to elasto-capillary fluid-solid interaction. International Journal for Numerical Methods in Engineering(March), 1–22. doi:10.1002/nme.6388