  We consider blocks of quantum spins in a chain at thermal equilibrium,
focusing on their properties from a thermodynamical perspective. Whereas in
classical systems the temperature behaves as an intensive magnitude, a
deviation from this behavior is expected in quantum systems. In particular, we
see that under some conditions the description of the blocks as thermal states
with the same global temperature as the whole chain fails. We analyze this
issue by employing the quantum fidelity as a figure of merit, singling out in
detail the departure from the classical behavior. The influence in this sense
of zero-temperature quantum phase transitions can be clearly observed within
this approach. Then we show that the blocks can be considered indeed as thermal
states with a high fidelity, provided an effective local temperature is
properly identified. Such a result originates from typical properties of
reduced sub-systems of energy-constrained Hilbert spaces. Finally, the relation
between local and global temperature is analyzed as a function of the size of
the blocks and the system parameters.
