The interplay of liquid-liquid phase separation (LLPS) with the glass transition is a potential route to the formation of arrested states in colloidal and protein systems. However, the details regarding the kinetic transition from a complete LLPS to an arrested state are still not clear. In our work, we have employed the USAXS technique to study the kinetics of LLPS and arrested state in protein systems with a high temporal and spatial resolution. The two protein systems studied were bovine γ-globulin in the presence of PEG1k, and bovine serum albumin with YCl3. For both systems, the growth kinetics of the characteristic length (ξ) for shallow quenches was found to follow the classical coarsening kinetics with ξ~t1/3. For deep quenches, a fully arrested state was obtained. Interestingly, for intermediate quenches, we observed an unusual three-stage coarsening process. We consider the state of the system following this type of kinetics as a temporarily arrested state. We interpret the finding in the light of simulations and experimental results on colloidal systems. The temporary nature of the arrested state is most likely due to the asymmetric quench. In this case, the minority phase is the dilute phase, which can still coarsen through the glassy majority phase.