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Sea-level rise may accelerate significantly if marine ice sheets become unstable. If such instability occurs, there would be considerable uncertainty in future sea-level rise projections due to imperfectly modeled ice sheet processes and unpredictable climate variability. In this study, we use mathematical and computational approaches to identify the ice sheet processes that drive uncertainty in sea-level projections. Using stochastic perturbation theory from statistical physics as a tool, we show mathematically that the marine ice sheet instability greatly amplifies and skews uncertainty in sea-level projections with worst-case scenarios of rapid sea-level rise being more likely than best-case scenarios of slower sea-level rise. We also perform large ensemble simulations with a state-of-the- art ice sheet model of Thwaites Glacier, a marine-terminating glacier in West Antarctica that is thought to be unstable. These ensemble simulations indicate that the uncertainty solely related to internal climate variability can be a large fraction of the total ice loss expected from Thwaites Glacier.We conclude that internal climate variability alone can be responsible for significant uncertainty in projections of sea-level rise and that large ensembles are a necessary tool for quantifying the upper bounds of this uncertainty.