  The spin-1/2 Ising-Heisenberg tetrahedral chain is exactly solved using its
local gauge symmetry, which enables one to establish a rigorous mapping with
the corresponding chain of composite Ising spins tractable within the
transfer-matrix approach. Exact results derived for spin-spin correlation
functions are employed to obtain the frustration temperature, at which a
product of correlation functions along an elementary triangular plaquette
becomes negative and the relevant spins experience a spin frustration. In
addition, we have exactly calculated a concurrence quantifying thermal
entanglement along with a threshold temperature, above which concurrence as a
measure of thermal entanglement vanishes. It is shown that the frustration and
threshold temperature coincide at sufficiently low temperatures, while they
exhibit a very different behavior in the high-temperature region when tending
towards completely different asymptotic limits. The threshold temperature
additionally shows a notable reentrant behavior when it extends over a narrow
temperature region above the classical ground state without any quantum
correlations. It is demonstrated that the specific heat may display temperature
dependence with or without an anomalous low-temperature peak for a relatively
strong or weak Heisenberg interaction, respectively.
