  We explore the notion of the quantum auxiliary linear problem and the
associated problem of quantum Backlund transformations (BT). In this context we
systematically construct the analogue of the classical formula that provides
the whole hierarchy of the time components of Lax pairs at the quantum level
for both closed and open integrable lattice models. The generic time evolution
operator formula is particularly interesting and novel at the quantum level
when dealing with systems with open boundary conditions. In the same frame we
show that the reflection K-matrix can also be viewed as a particular type of
BT, fixed at the boundaries of the system. The q-oscillator (q-boson) model, a
variant of the Ablowitz-Ladik model, is then employed as a paradigm to
illustrate the method. Particular emphasis is given to the time part of the
quantum BT as possible connections and applications to the problem of quantum
quenches as well as the time evolution of local quantum impurities are evident.
A discussion on the use of Bethe states as well as coherent states and the path
integral formulation for the study of the time evolution is also presented.
