  The persistent separability of certain quantum states, known as symmetric
absolutely separable (SAS), under symmetry-preserving global unitary
transformations is of key significance in the context of quantum resources for
bosonic systems. In this work, we develop criteria for detecting SAS states of
any number of qubits. Our approach is based on the Glauber-Sudarshan $P$
representation for finite-dimensional quantum systems. We introduce three
families of SAS witnesses, one linear and two nonlinear in the eigenvalues of
the state, formulated respectively as an algebraic inequality or a quadratic
optimization problem. These witnesses are capable of identifying more SAS
states than previously known counterparts. We also explore the geometric
properties of the subsets of SAS states detected by our witnesses, shedding
light on their distinctions.
