  I introduce a new notion, that extends the mutually unbiased bases (MUB)
conditons to more than two bases. These, I call the nUB conditions, and the
corresponding bases $n$-fold unbiased. They naturally appear while optimizing
generic $n$-to-one quantum random access code (QRAC) strategies. While their
existence in general dimensions is an open question, they nevertheless give
close-to-tight upper bounds on QRAC success probabilities, and raise
fundamental questions about the geometry of quantum states.
