Understanding the relationship between efficient quantum and classical computation is one of the major goals of quantum complexity theory. To this end we have considered some natural classes of quantum computations and compared their strength and their properties to those of the corresponding classical complexity classes. In this talk we focus on two examples of this work where the quantum and classical worlds differ. We consider a natural (and strong) quantum analog of NP and show it is equivalent to a to a counting class which is hard for the polynomial hierarchy, and we consider the class of quantum AC0 circuits, and in the presence of unitary fanout gates, prove that adding mod gates (for example parity gates) does not increase their computational power.