Explicit Chabauty-Kim theory for the thrice punctured line in depth 2

Let X = P₁ / {0, 1,∞}, and let S denote a finite set of prime numbers. In an article of 2005, Kim gave a new proof of Siegel's theorem for X: the set X(Z[S^-1]) of S-integral points of X is finite. The proof relies on a 'nonabelian' version of the classical Chabauty method. At its heart is a modular interpretation of unipotent p-adic Hodge theory, given by a tower of morphisms hn between certain Q_p-varieties. We set out to obtain a better understanding of h₂. Its mysterious piece is a polynomial in 2|S| variables. Our main theorem states that this polynomial is quadratic, and gives a procedure for writing its coefficients in terms of p-adic logarithms and dilogarithms.