Emergence of strong-coupling superconductivity and quantum criticality correlated with Lifshitz transitions in the alternate stacking compound 4Hb−TaS2

$4{H}_{b}\text{\ensuremath{-}}\mathrm{Ta}{\mathrm{S}}_{2}$ is a naturally formed quasi-two-dimensional heterojunction material composed of alternating monolayers of insulating ($1T$-) and superconducting ($1H$-) $\mathrm{Ta}{\mathrm{S}}_{2}$. We report on a comprehensive high-pressure study on the interplay between charge-density wave (CDW) and superconductivity (SC) in $4{H}_{b}\text{\ensuremath{-}}\mathrm{Ta}{\mathrm{S}}_{2}$. The results uncover a dome-shaped strong-coupling superconductor on the border of a 3 \ifmmode\times\else\texttimes\fi{} 3 commensurate CDW at ${P}_{\mathrm{c}1}\ensuremath{\approx}2.0$ GPa: (1) nearly one order enhancement of upper critical field ${B}_{c2}$(0); (2) the derived $2{\mathrm{\ensuremath{\Delta}}}_{0}/{k}_{\mathrm{B}}{T}_{c}$ beyond the BCS theory, decaying to a conventional one above 4.5 GPa; (3) the exponent of normal-state resistivity $n\ensuremath{\approx}1.50$ and triply enhanced electronic effective mass. Under pressure, a third CDW emerging from 2.0 GPa is related to the original two CDWs, then disappears above 11.5 GPa. The temperature dependence of ${B}_{c2}(T)$ collapses into a universal curve and the comparison of ${B}_{c2}(T)$ to a polar-state function in $4{H}_{b}\text{\ensuremath{-}}\mathrm{Ta}{\mathrm{S}}_{2}$. Theoretical calculations proposed the stronger interlayer coupling and band hybridization responsible for strong-coupling SC; the suppression of new CDWs is related to band inversions between the $1T$-Ta-${d}_{xy}$ (GM1+) and S-${p}_{z}$ bands (GM2-) at the $\mathrm{\ensuremath{\Gamma}}A$ point; above 5.0 GPa, the coexisting weak-coupling SC and linear magnetoresistance can be attributed to the formation of electronic bands along the $M\ensuremath{-}K$ lines and the $\mathrm{\ensuremath{\Gamma}}A$ point. Our discovery provides an excellent example to demonstrate the interplay of the strong-coupling superconducting state and topological electronic state in van der Waals heterojunctions.