  The discrete truncated Wigner approximation (DTWA) is a powerful tool for
analyzing dynamics of quantum spin systems. Since the DTWA includes the
leading-order quantum corrections to a mean-field approximation, it is
naturally expected that the DTWA becomes more accurate when the range of
interactions of the system increases. However, quantitative corroboration of
this expectation is still lacking mainly because it is generally difficult in a
large system to evaluate a timescale on which the DTWA is quantitatively valid.
In order to investigate how the validity timescale depends on the interaction
range, we analyze dynamics of quantum spin models with a step function type
interaction subjected to a sudden quench of a magnetic field by means of both
DTWA and its extension including the second-order correction, which is derived
from the Bogoliubov-Born-Green-Kirkwood-Yvon equation. We also develop a
formulation for calculating the second-order R\'enyi entropy within the
framework of the DTWA. By comparing the time evolution of the R\'enyi entropy
computed by the DTWA with that by the extension including the correction, we
find that both in the one- and two-dimensional systems the validity timescale
increases algebraically with the range of the step function type interaction.
