  By using the quantum-memory-assisted entropic uncertainty relation (EUR), we
derive a computable tight upper bound for quantum discord, which applies to an
arbitrary bipartite state. Detailed examples show that this upper bound is
tighter than other known bounds in a wide regime. Furthermore, we show that for
any tripartite pure state, the quantum-memory-assisted EUR imposes a constraint
on the shareability of quantum correlations among the constituent parties. This
conclusion amends the well accepted result that quantum discord is not
monogamous.
