  We develop methods for calculating the zero-frequency noise for quantum
shuttles, i.e. nanoelectromechanical devices where the mechanical motion is
quantized. As a model system we consider a three-dot array, where the internal
electronic coherence both complicates and enriches the physics. Two different
formulations are presented: (i) quantum regression theorem, and (ii) the
counting variable approach. It is demonstrated, both analytically and
numerically, that the two formulations yield identical results, when the
conditions of their respective applicability are fulfilled. We describe the
results of extensive numerical calculations for current and current noise (Fano
factor), based on a solution of a Markovian generalized master equation. The
results for the current and noise are further analyzed in terms of Wigner
functions, which help to distinguish different transport regimes (in
particular, shuttling vs. cotunneling). In the case of weak inter-dot coupling,
the electron transport proceeds via sequential tunneling between neighboring
dots. A simple rate equation with the rates calculated analytically from the
P(E)-theory is developed and shown to agree with the full numerics.
