  Context. Investigation of the hard X-ray emission properties of blazars is
key to the understanding of the central engine of the sources and associated
jet process. In particular, simultaneous spectral and timing analyses of the
intra-day hard X-ray observations provide us a means to peer into the compact
innermost blazar regions, not accessible to our current instruments. Aims. The
primary objective of the work is to associate the observed hard X-ray
variability properties in the blazars to their flux and spectral states,
thereby, based on the correlation among them, extract the details about the
emission regions and the processes occurring near the central engine. Methods.
We carried out timing, spectral and cross-correlation analysis of 31 NuSTAR
observations of 13 blazars. We investigated the spectral shapes of the sources
using single power-law, broken power-law and log-parabola models. We also
studies the co-relation between the soft and the hard emission using
z-transformed discrete correlation function. Results. We found that for most of
the sources the hard X-ray emission can be well represented by log-parabola
model; and that the spectral slopes for different blazar sub-classes are
consistent with so called blazar sequence. We noted a close connection between
the flux and spectral slope within the source sub-class in the sense that high
flux and/or flux states tend to be harder in spectra. In BL Lacertae objects,
assuming particle acceleration by diffusive shocks and synchrotron cooling as
the dominant processes governing the observed flux variability, we constrain
the magnetic field of the emission region to be a few gauss; whereas in
flat-spectrum radio quasars, using external Compton models, we estimate the
energy of the lower end of the injected electrons to be a few Lorentz factors.
