  In this paper we study various aspects of the double ramification (DR)
hierarchy, introduced by the first author, and its quantization. We extend the
notion of tau-symmetry to quantum integrable hierarchies and prove that the
quantum DR hierarchy enjoys this property. We determine explicitly the genus
$1$ quantum correction and, as an application, compute completely the
quantization of the $3$- and $4$-KdV hierarchies (the DR hierarchies for
Witten's $3$- and $4$-spin theories). We then focus on the recursion relation
satisfied by the DR Hamiltonian densities and, abstracting from its geometric
origin, we use it to characterize and construct a new family of quantum and
classical integrable systems which we call of double ramification type, as they
satisfy all of the main properties of the DR hierarchy. In the second part, we
obtain new insight towards the Miura equivalence conjecture between the DR and
Dubrovin-Zhang hierarchies, via a geometric interpretation of the correlators
forming the double ramification tau-function. We then show that the candidate
Miura transformation between the DR and DZ hierarchies (which we uniquely
identified in our previous paper) indeed turns the Dubrovin-Zhang Poisson
structure into the standard form. Eventually, we focus on integrable
hierarchies associated with rank-$1$ cohomological field theories and their
deformations, and we prove the DR/DZ equivalence conjecture up to genus $5$ in
this context.
