  We identify five selected open problems in the theory of quantum information,
which are rather simple to formulate, were well-studied in the literature, but
are technically not easy. As these problems enjoy diverse mathematical
connections, they offer a huge breakthrough potential. The first four concern
existence of certain objects relevant for quantum information, namely a family
of symmetric informationally complete generalized measurements in an infinite
sequence of dimensions, mutually unbiased bases in dimension six, absolutely
maximally entangled states for four subsystems with six levels each and bound
entangled states with negative partial transpose. The fifth problem requires
checking whether a certain state of a two-ququart system is 2-copy distillable.
An award for solving each of them is announced.
