  Efficiently measuring nonlinear properties, like the entanglement spectrum,
is a significant yet challenging task from quantum information processing to
many-body physics. Current methodologies often suffer from an exponential
scaling of the sampling cost or require auxiliary qubits and deep quantum
circuits. To address these limitations, we propose an efficient auxiliary-free
replica shadow (AFRS) framework, which leverages the power of the joint
entangling operation on a few input replicas while integrating the mindset of
shadow estimation. We rigorously prove that AFRS can offer exponential
improvements in estimation accuracy compared with the conventional shadow
method, and facilitate the simultaneous estimation of various nonlinear
properties, unlike the destructive swap test. Additionally, we introduce an
advanced local-AFRS variant tailored to estimating local observables with even
constant-depth local quantum circuits, which significantly simplifies the
experimental realization compared with the general swap test. Our work paves
the way for the application of AFRS on near-term quantum hardware, opening new
avenues for efficient and practical quantum measurements.
