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Multilevel Cell Spin-Transfer Torque Random Access Memory (MLC STT-RAM) is a promising nonvolatile memory technology to build registers for its natural immunity to electromagnetic radiation in rad-hard space environment. Unlike traditional SRAM-based registers, MLC STT-RAM exhibits unbalanced write state transitions due to the fact that the magnetization directions of hard and soft domains cannot be flipped independently. This feature leads to nonuniform costs of write states in terms of latency and energy. However, current SRAM-targeting register allocations do not have a clear understanding of the impact of the different write state-transition costs. As a result, those approaches heuristically select variables to be spilled without considering the spilling priority imposed by MLC STT-RAM. Aiming to address this limitation, this paper proposes a state-transition-aware spilling cost minimization (SSCM) policy, to save power when MLC STT-RAM is employed in register design. Specifically, the spilling cost model is first constructed according to the linear combination of different state-transition frequencies. Directed by the proposed cost model, the compiler picks up spilling candidates to achieve lower power and higher performance. Experimental results show that the proposed SSCM technique can save energy by 19.4% and improve the lifetime by 23.2% of MLC STT-RAM-based register design.