  We propose a protocol to encode classical bits in the measurement statistics
of many-body Pauli observables, leveraging quantum correlations for a random
access code. Measurement contexts built with these observables yield outcomes
with intrinsic redundancy, something we exploit by encoding the data into a set
of convenient context eigenstates. This allows to randomly access the encoded
data with few resources. The eigenstates used are highly entangled and can be
generated by a discretely-parametrized quantum circuit of low depth.
Applications of this protocol include algorithms requiring large-data storage
with only partial retrieval, as is the case of decision trees. Using $n$-qubit
states, this Quantum Random Access Code has greater success probability than
its classical counterpart for $n\ge 14$ and than previous Quantum Random Access
Codes for $n \ge 16$. Furthermore, for $n\ge 18$, it can be amplified into a
nearly-lossless compression protocol with success probability $0.999$ and
compression ratio $O(n^2/2^n)$. The data it can store is equal to Google-Drive
server capacity for $n= 44$, and to a brute-force solution for chess (what to
do on any board configuration) for $n= 100$.
