  We present the first results of a search for transient hard X-ray (HXR)
emission in the quiet solar corona with the \textit{Nuclear Spectroscopic
Telescope Array} (\textit{NuSTAR}) satellite. While \textit{NuSTAR} was
designed as an astrophysics mission, it can observe the Sun above 2~keV with
unprecedented sensitivity due to its pioneering use of focusing optics.
\textit{NuSTAR} first observed quiet Sun regions on 2014 November 1, although
out-of-view active regions contributed a notable amount of background in the
form of single-bounce (unfocused) X-rays. We conducted a search for quiet Sun
transient brightenings on time scales of 100 s and set upper limits on emission
in two energy bands. We set 2.5--4~keV limits on brightenings with time scales
of 100 s, expressed as the temperature T and emission measure EM of a thermal
plasma. We also set 10--20~keV limits on brightenings with time scales of 30,
60, and 100 s, expressed as model-independent photon fluxes. The limits in both
bands are well below previous HXR microflare detections, though not low enough
to detect events of equivalent T and EM as quiet Sun brightenings seen in soft
X-ray observations. We expect future observations during solar minimum to
increase the \textit{NuSTAR} sensitivity by over two orders of magnitude due to
higher instrument livetime and reduced solar background.
