  Neutrino non-standard interactions (NSI) can be constrained using coherent
elastic neutrino-nucleus scattering. We discuss here two aspects in this
respect, namely the effects of (i) charged current NSI in neutrino production
and (ii) CP-violating phases associated with neutral current NSI in neutrino
detection. Effects of CP-phases require the simultaneous presence of two
different flavor-changing neutral current NSI parameters. Applying these two
scenarios to the COHERENT measurement, we derive limits on charged current NSI
and find that more data is required to compete with the existing limits.
Regarding CP-phases, we show how the limits on the NSI parameters depend
dramatically on the values of the phases. Accidentally, the same parameters
influencing coherent scattering also show up in neutrino oscillation
experiments. We find that COHERENT provides complementary constraints on the
set of NSI parameters that can explain the discrepancy in the best-fit value of
the standard CP-phase obtained by T2K and NO$\nu$A, while the significance with
which the LMA-Dark solution is ruled out can be weakened by the presence of
additional NSI parameters introduced here.
