I will talk about the proofs of two formulas stated by Dale Peterson which compute the structure coefficients of the small quantum cohomology ring of G/P. The first is a formula for multiplication by a divisor class, and is a simple application of the Bott residue formula and Kleiman-Bertini in the moduli space of stable maps. This formula is sufficient to compute all the small GW invariants for G/B. The second is a replacement for functoriality: it expresses any small GW invariant of G/P in terms of one for G/B, and can be proved by studying the type of the Harder-Narasimhan and Jordan-Holder filtrations of the pull-back of the tautological bundle over G/P, under a general stable map.