  Quantum entanglement can provide enhanced precision beyond standard quantum
limit (SQL), the highest precision achievable with classical means. It remains
challenging, however, to observe large enhancement limited by the experimental
abilities to prepare, maintain, manipulate and detect entanglement. Here, we
present nonlinear interferometry protocols based on echoing spin-nematic
squeezing to achieve record high enhancement factors in atomic Bose-Einstein
condensate. The echo is realized by a state-flip of the spin-nematic squeezed
vacuum, which serves as the probe state and is refocused back to the vicinity
of the unsqueezed initial state while carrying out near noiseless amplification
of a signal encoded. A sensitivity of $21.6\pm0.5$ decibels (dB) for a
small-angle Rabi rotation beyond the two-mode SQL of 26400 atoms as well as
$16.6\pm1.3$ dB for phase sensing in a Ramsey interferometer are observed. The
absolute phase sensitivity for the latter extrapolates to
$103~\rm{pT/\sqrt{Hz}}$ at a probe volume of $18~\mu\rm{m}^3$ for near-resonant
microwave field sensing. Our work highlights the excellent many-body coherence
of spin-nematic squeezing and suggests its possible quantum metrological
applications in atomic magnetometer, atomic optical clock, and fundamental
testing of Lorentz symmetry violation, etc.
