Can quantum control modify thermodynamic behavior?

We review the effects of frequent, impulsive quantum nondemolition measurements of the energy of two-level systems, alias qubits, in contact with a thermal bath. The resulting entropy and temperature of the system subject to measurements at intervals below the bath memory (Markovianity) time are completely determined by the measurement rate. Namely, they are unrelated to what is expected by standard thermodynamical behavior that holds for Markovian baths. These anomalies allow for very fast control of heating, cooling, and state-purification (entropy reduction) of qubits, much sooner than their thermal equilibration time. We further show that frequent measurements may enable the extraction of work in a closed cycle from the system-bath interaction (correlation) energy, a hitherto unexploited work resource. They allow for work even if no information is gathered or the bath is at zero temperature, provided the cycle is within the bath memory time.