  We introduce a method that allows one to infer many properties of a quantum
state -- including nonlinear functions such as R\'enyi entropies -- using only
global control over the constituent degrees of freedom. In this protocol, the
state of interest is first entangled with a set of ancillas under a fixed
global unitary, before projective measurements are made. We show that when the
unitary is sufficiently entangling, a universal relationship between the
statistics of the measurement outcomes and properties of the state emerges,
which can be connected to the recently discovered phenomenon of emergent
quantum state designs in chaotic systems. Thanks to this relationship,
arbitrary observables can be reconstructed using the same number of
experimental repetitions that would be required in classical shadow tomography
[Huang et al., Nat. Phys. 16, 1050 (2020)]. Unlike previous approaches to
shadow tomography, our protocol can be implemented using only global
operations, as opposed to qubit-selective logic gates, which makes it
particularly well-suited to analog quantum simulators, including ultracold
atoms in optical lattices and arrays of Rydberg atoms.
