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We report magnetocaloric and magnetic-torque evidence that in —a geometrically frustrated Heisenberg triangular-lattice antiferromagnet—quantum fluctuations stabilize a series of spin states at simple increasing fractions of the saturation magnetization . Only the first of these states—at —has been theoretically predicted. We discuss how the higher fraction quantum states might arise and propose model spin arrangements. We argue that the first-order nature of the transitions into those states is due to strong lowering of the energies by quantum fluctuations, with implications for the general character of quantum phase transitions in geometrically frustrated systems.