  The identification of spin-nematic states is challenging due to the absence
of Bragg peaks. However, the study of dynamical physical quantities provides a
promising avenue for characterizing these states. In this study, we investigate
the dynamical properties of spin-nematic states in three-dimensional quantum
spin systems in a magnetic field, using a two-component boson theory that
incorporates magnons and bi-magnons. Our particular focus lies on the dynamical
spin structure factor at zero temperature and the nuclear magnetic resonance
(NMR) relaxation rate at finite temperatures. Our findings reveal that the
dynamical structure factor does not exhibit any diverging singularity across
momentum and frequency while providing valuable information about the form
factor of bi-magnon states and the underlying structure of spin-nematic order.
Furthermore, we find a temperature dependence in the NMR relaxation rate
proportional to $T^3$ at low temperatures, similar to canted antiferromagnets.
A clear distinction arises as there is no critical divergence of the NMR
relaxation rate at the spin-nematic transition temperature. Our theoretical
framework provides a comprehensive understanding of the excitation spectrum and
the dynamical properties of spin-nematic states, covering arbitrary spin values
$S$ and encompassing site and bond nematic orders. Additionally, we apply the
same methodology to analyze these dynamical quantities in a canted
antiferromagnetic state and compare the results with those in the spin-nematic
states.
