  There are at least three models of discrete-time quantum walks (QWs) on
graphs currently under active development. In this work we focus on the
equivalence of two of them, known as Szegedy's and staggered QWs. We give a
formal definition of the staggered model and discuss generalized versions for
searching marked vertices. Using this formal definition, we prove that any
instance of Szegedy's model is equivalent to an instance of the staggered
model. On the other hand, we show that there are instances of the staggered
model that cannot be cast into Szegedy's framework. Our analysis also works
when there are marked vertices. We show that Szegedy's spatial search
algorithms can be converted into search algorithms in staggered QWs. We take
advantage of the similarity of those models to define the quantum hitting time
in the staggered model and to describe a method to calculate the eigenvalues
and eigenvectors of the evolution operator of staggered QWs.
