Median treatment effect in randomized trials

For two response variables y(t) and y(c) corresponding to two treatments (or two policies) T and C, we wish to learn about quantiles of y(t) - y(c) from the marginal quantiles of y(t) and y(c); only one of y(t) and y(c) is observed for an individual. We find that, in general, this is difficult for quantiles other than the median unless strong assumptions are imposed on how y(t) is related to y(c). For the median, we present conditions under which the sign of the median treatment effect is identified.