  We consider two classes of quantum generalisations of Random Access Code
(RAC) and study lower bounds for probabilities of success for such tasks. It
provides a useful framework for the study of certain information processing
tasks with constrained resources. The first class is based on a random access
code with quantum inputs and output known as No-Signalling Quantum RAC
(NS-QRAC) [A. Grudka et al. Phys. Rev. A 92, 052312 (2015)], where unbounded
entanglement and constrained classical communication are allowed, which can be
seen as quantum teleportation with constrained classical communication, for
which we provide a quantum lower bound. We consider two modifications to the
NS-QRAC scenario, first where unbounded entanglement and constrained quantum
communication is allowed and, second where bounded entanglement and
unconstrained classical communication are allowed, where we find a monogamy
relation for the transmission fidelities, which -- in contrast to the usual
communication schemes -- involves multiple senders and a single receiver. We
provide lower bounds for these scenarios. The second class is based on a random
access code with a quantum channel and shared entanglement [A. Tavakoli et al.
PRX Quantum 2 (4) 040357 (2021)]. We study the set of tasks where two inputs
made of two digits of $d$-base are encoded over a qudit and a maximally
entangled state, which can be seen as quantum dense coding with constrained
quantum communication, for which we provide quantum lower bounds for $d=2,3,4$.
The encoding employed utilises Gray codes.
