We report heat capacity measurements of SrCu$_2$(BO$_3$)$_2$ under high pressure along with simulations of relevant quantum spin models and map out the $(P,T)$ phase diagram of the material. We find a first-order quantum phase transition between the low-pressure quantum dimer paramagnet and a phase with signatures of a plaquette-singlet state below T = $2$ K. At higher pressures, we observe a transition into a previously unknown antiferromagnetic state below $4$ K. Our findings can be explained within the two-dimensional Shastry-Sutherland quantum spin model supplemented by weak inter-layer couplings. The possibility to tune SrCu$_2$(BO$_3$)$_2$ between the plaquette-singlet and antiferromagnetic states opens opportunities for experimental tests of quantum field theories and lattice models involving fractionalized excitations, emergent symmetries, and gauge fluctuations.
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