$n$-fold unbiased bases: an extension of the MUB condition

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.