  We provide a complete characterisation of the Root solution to the Skorokhod
embedding problem (SEP) by means of an optimal stopping formulation. Our
methods are purely probabilistic and the analysis relies on a tailored
time-reversal argument. This approach allows us to address the long-standing
question of a multiple marginals extension of the Root solution of the SEP. Our
main result establishes a complete solution to the n-marginal SEP using first
hitting times of barrier sets by the time-space process. The barriers are
characterised by means of a recursive sequence of optimal stopping problems.
Moreover, we prove that our solution enjoys a global optimality property
extending the one-marginal Root case. Our results hold for general,
one-dimensional, martingale diffusions.
