I will discuss the multimarginal optimal transport problem with Coulomb cost as it arises in quantum chemistry and statistical physics. I will highlight the current state of knowledge and the open questions surrounding the structure and regularity of the minimizer(s) to this problem for which not much is known. I will then present some results of mine on the Kantorovich duality, notably a key property of the so-called Kantorovich potential that I use to construct a physically relevant numerical discretization of the problem.