  We discover that quantum dynamical tunneling, occurring between phase space
regions in a classically forbidden way, can break conserved quantities in
pseudointegrable systems. We rigorously prove that a conserved quantity in a
class of typical pseudointegrable systems can be broken quantum mechanically.
We then numerically compute the uncertainties of this broken conserved
quantity, which remain non-zero for up to $10^5$ eigenstates and exhibit
universal distributions similar to energy level statistics. Furthermore, all
the eigenstates with large uncertainties show the superpositions of regular
orbits with different values of the conserved quantity, showing definitive
manifestation of dynamical tunneling. A random matrix model is constructed to
successfully reproduce the level statistics in pseudointegrable systems.
