| Bloch's conjecture for a surface X over an algebraically closed field k states that every homologically trivial correspondence acts trivially on the Albanese kernel of the extension of X to a universal domain containing k. Here we prove that, for a complex K3 surface X, Bloch's conjecture is equivalent to the existence of a valence for every correspondence. We also give applications of this result to the case of a correspondence associated to an automorphism of finite order and to the existence of constant cycle curves on X. Finally we show that Franchetta's conjecture, as stated by K. O'Grady, holds true for the family of polarized K3 surfaces of genus g, if g is greater or equal to 3 and smaller or equal to 6. |