  We investigate the quantitative constraint on the triple-alpha reaction rate
based on stellar evolution theory, motivated by the recent significant revision
of the rate proposed by nuclear physics calculations. Targeted stellar models
were computed in order to investigate the impact of that rate in the mass range
of 0.8 < M / Msun < 25 and in the metallicity range between Z = 0 and Z = 0.02.
The revised rate has a significant impact on the evolution of low- and
intermediate-mass stars, while its influence on the evolution of massive stars
(M >~ 10 Msun) is minimal. We find that employing the revised rate suppresses
helium shell flashes on AGB phase for stars in the initial mass range 0.8 < M /
Msun < 6, which is contradictory to what is observed. The absence of helium
shell flashes is due to the weak temperature dependence of the revised
triple-alpha reaction cross section at the temperature involved. In our models,
it is suggested that the temperature dependence of the cross section should
have at least nu > 10 at T = 1 - 1.2 x 10^8 K where the cross section is
proportional to T^{nu}. We also derive the helium ignition curve to estimate
the maximum cross section to retain the low-mass first red giants. The
semi-analytically derived ignition curves suggest that the reaction rate should
be less than ~ 10^{-29} cm^6 s^{-1} mole^{-2} at ~ 10^{7.8} K, which
corresponds to about three orders of magnitude larger than that of the NACRE
compilation. In an effort to compromise with the revised rates, we calculate
and analyze models with enhanced CNO cycle reaction rates to increase the
maximum luminosity of the first giant branch. However, it is impossible to
reach the typical RGB tip luminosity even if all the reaction rates related to
CNO cycles are enhanced by more than ten orders of magnitude.
