In biological systems, prolonged conflict is costly, whereas contained conflict permits strategic innovation and refinement. Causes of variation in conflict size and duration are not well understood. We use a well-studied primate society model system to study how conflicts grow. We find conflict duration is a ‘first to fight’ growth process that scales superlinearly with the number of possible pairwise interactions. This is in contrast with a ‘first to fail’ process that characterizes peaceful durations. Rescaling conflict distributions reveals a universal curve, showing that the typical timescale of correlated interactions exceeds nearly all individual fights. This temporal correlation implies collective memory across pairwise interactions beyond those assumed in the standard models of contagion growth or iterated evolutionary games. By accounting for memory, we make quantitative predictions for interventions that mitigate or enhance the spread of conflict. Managing conflict involves balancing the efficient use of limited resources with an intervention strategy that allows for conflict, but keeps it contained and controlled.