Spectroscopy and lifetime measurements in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi>Te</mml:mi><mml:mprescripts/><mml:none/><mml:mrow><mml:mn>134</mml:mn><mml:mo>,</mml:mo><mml:mn>136</mml:mn><mml:mo>,</mml:mo><mml:mn>138</mml:mn></mml:mrow></mml:mmultiscripts></mml:math> isotopes and implications for the nuclear structure beyond <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>82</mml:mn></mml:mrow></mml:math>

We report on spectroscopic information and lifetime measurements of even-even neutron-rich Te isotopes. Excited states were populated in fast-neutron induced fission of $^{238}\mathrm{U}$ at the ALTO facility of IJCLab with the LICORNE neutron source and detected using the hybrid $\ensuremath{\nu}$-ball spectrometer. We provide first results on lifetimes of the ${6}_{1}^{+}$ state in $^{136}\mathrm{Te}$ and the (${6}_{1}^{+}$), (${4}_{1}^{+}$), and (${2}_{1}^{+}$) states in $^{138}\mathrm{Te}$ and discuss the results in the context of large-scale shell-model calculations. The level schemes of $^{136}\mathrm{Te}$ and $^{138}\mathrm{Te}$ are revised in terms of lifetimes of their ${2}_{1}^{+},{4}_{1}^{+}$ states and updated information on the $({4}_{2}^{+})$ state in $^{136}\mathrm{Te}$ is presented. In addition, previously reported data on spectroscopy and lifetimes in $^{134}\mathrm{Te}$ are reexamined. This work provides new insights into the evolution of collectivity for Te isotopes with $N=82,84,86$.