A bstract We analyze spin entanglement in $\Lambda \overline{\Lambda}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miΛ</mml:mi> mml:mover mml:miΛ</mml:mi> mml:mo¯</mml:mo> </mml:mover> </mml:math> pairs produced in the decays of spin-zero particles, contrasting predictions from quantum field theory (QFT) with those of local hidden-variable theories (LHVTs). Using the self-analyzing weak decays Λ → pπ − and $\overline{\Lambda}\to \overline{p}{\pi}^{+}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mover mml:miΛ</mml:mi> mml:mo¯</mml:mo> </mml:mover> mml:mo→</mml:mo> mml:mover mml:mip</mml:mi> mml:mo¯</mml:mo> </mml:mover> mml:msup mml:miπ</mml:mi> mml:mo+</mml:mo> </mml:msup> </mml:math> , we derive the joint angular distributions within QFT. Our key findings are: for scalar production $h\to \Lambda \overline{\Lambda}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mih</mml:mi> mml:mo→</mml:mo> mml:miΛ</mml:mi> mml:mover mml:miΛ</mml:mi> mml:mo¯</mml:mo> </mml:mover> </mml:math> , no LHVT respecting locality and angular-momentum conservation can reproduce the QFT distribution. For pseudoscalar production $a\to \Lambda \overline{\Lambda}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mia</mml:mi> mml:mo→</mml:mo> mml:miΛ</mml:mi> mml:mover mml:miΛ</mml:mi> mml:mo¯</mml:mo> </mml:mover> </mml:math> , a CPT-symmetric LHVT is excluded by positivity constraints given the measured analyzing powers; however, if CPT symmetry is relaxed, an explicit LHVT construction — with uniform hidden-variable measure and response functions satisfying ${b}_1{c}_1=3{\alpha}_{\Lambda}{\alpha}_{\overline{\Lambda}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msub mml:mib</mml:mi> mml:mn1</mml:mn> </mml:msub> mml:msub mml:mic</mml:mi> mml:mn1</mml:mn> </mml:msub> mml:mo=</mml:mo> mml:mn3</mml:mn> mml:msub mml:miα</mml:mi> mml:miΛ</mml:mi> </mml:msub> mml:msub mml:miα</mml:mi> mml:mover mml:miΛ</mml:mi> mml:mo¯</mml:mo> </mml:mover> </mml:msub> </mml:math> — can match the QFT result. For the most general spin-zero decay $s\to \Lambda \overline{\Lambda}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mis</mml:mi> mml:mo→</mml:mo> mml:miΛ</mml:mi> mml:mover mml:miΛ</mml:mi> mml:mo¯</mml:mo> </mml:mover> </mml:math> with arbitrary scalar-pseudoscalar mixing, we, under CPT invariance, identify the regions of parameter space where the QFT joint angular distribution does or does not admit an LHVT realization. These distinct signatures provide clear, experimentally testable criteria to discriminate between QFT and LHVT in $\Lambda \overline{\Lambda}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miΛ</mml:mi> mml:mover mml:miΛ</mml:mi> mml:mo¯</mml:mo> </mml:mover> </mml:math> systems across different production mechanisms.