  We introduce an even-parity precession protocol that can detect
nonclassicality of some quantum states using only measurements of a
uniformly-precessing variable at different points in time. Depending on the
system under study, the protocol may detect the Wigner negativity of a single
quantum harmonic oscillator or of a single spin $j\geq 2$; the non-Gaussian
entanglement of two harmonic oscillators; or genuine multipartite entanglement
of a spin ensemble, whose total spin is integer. Unlike other nonclassicality
tests, simultaneous or sequential measurements are not required. Our protocol
can also detect states that commute with the parity operator, which were missed
by similar protocols built from Tsirelson's original precession protocol. This
work also closes a long-standing gap by showing the possibility of detecting
the Greenberger--Horne--Zeilinger entanglement of an even number of qubits
using only collective spin measurements.
