  The Third Law of Thermodynamics constrains the phase diagram of{systems with
a first-order quantum phase transition. For zero conjugate field, the
coexistence curve has an infinite slope at T=0. If a tricritical point exists
at T>0, then the associated tricritical wings are perpendicular to the T=0
plane, but not to the zero-field plane. These results are based on the third
law and basic thermodynamics only, and are completely general. As an explicit
example we consider the ferromagnetic quantum phase transition in clean metals,
where a first-order quantum phase transition is commonly observed.
