  It is anticipated that the gravitational radiation detected in future
gravitational wave (GW) detectors from binary neutron star (NS) mergers can
probe the high-density equation of state (EOS). We perform the first
simulations of binary NS mergers which adopt various parametrizations of the
quark-hadron crossover (QHC) EOS. These are constructed from combinations of a
hadronic EOS ($n_{b} < 2~n_0$) and a quark-matter EOS ($n_{b} >~5~n_0$), where
$n_{b}$ and $n_0$ are the baryon number density and the nuclear saturation
density, respectively. At the crossover densities ($2~ n_0 < n_{b} < 5~ n_0$)
the QHC EOSs continuously soften, while remaining stiffer than hadronic and
first-order phase transition EOSs, achieving the stiffness of strongly
correlated quark matter. This enhanced stiffness leads to significantly longer
lifetimes of the postmerger NS than that for a pure hadronic EOS. We find a
dual nature of these EOSs such that their maximum chirp GW frequencies
$f_{max}$ fall into the category of a soft EOS while the dominant peak
frequencies ($f_{peak}$) of the postmerger stage fall in between that of a soft
and stiff hadronic EOS. An observation of this kind of dual nature in the
characteristic GW frequencies will provide crucial evidence for the existence
of strongly interacting quark matter at the crossover densities for QCD.
