  We give sufficient conditions for the semisimplicity of quantum cohomology of
Fano varieties of Picard rank 1. We apply these techniques to prove new
semisimplicity results for some Fano varieties of Picard rank 1 and large
index. We also give examples of Fano varieties having a non semisimple small
quantum cohomology but a semisimple big quantum cohomology.
