  In this paper, we investigate the relationship between the quantum speedup,
nonMarkovianity and formation of a system-environment bound state. Previous
results show a monotonic relation between these three such that providing
stronger bound states can lead to a higher degree of non-Markovianity, and
hence to greater speed of quantum evolution. By studying dynamics of a
dissipative two-level system or a V-type three-level system in the presence of
similar and additional systems, we reveal that the quantum speedup is
exclusively related to the formation of the system-environment bound state,
while the non-Markovian effect of the system dynamics is neither necessary nor
sufficient to speed up the quantum evolution. In the other hand, it is
demonstrated that only the formation of the system-environment bound state
plays a decisive role in the acceleration of the quantum evolution.
