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We study the zero-temperature phase diagram of the 2D quantum$J_{1}^{XXZ}$--$J_{2}^{XXZ}$ spin-1/2 anisotropic Heisenberg model on thesquare lattice. In particular, the effects of the anisotropy $\Delta$ on the$z$-aligned N\'{e}el and (collinear) stripe states, as well as on the$xy$-planar-aligned N\'{e}el and collinear stripe states, are examined. Allfour of these quasiclassical states are chosen in turn as model states on topof which we systematically include the quantum correlations using a coupledcluster method analysis carried out to very high orders. We find strongevidence for two {\it quantum triple points} (QTP's) at ($\Delta ^{c} = -0.10\pm 0.15, J_{2}^{c}/J_{1} = 0.505 \pm 0.015$) and ($\Delta ^{c} = 2.05 \pm0.15, J_{2}^{c}/J_{1} = 0.530 \pm 0.015$), between which an intermediatemagnetically-disordered phase emerges to separate the quasiclassical N\'{e}eland stripe collinear phases. Above the upper QTP ($\Delta \gtrsim 2.0$) we finda direct first-order phase transition between the N\'{e}el and stripe phases,exactly as for the classical case. The $z$-aligned and $xy$-planar-alignedphases meet precisely at $\Delta = 1$, also as for the classical case. For allvalues of the anisotropy parameter between those of the two QTP's there existsa narrow range of values of $J_{2}/J_{1}$, $\alpha^{c_1}(\Delta)<J_{2}/J_{1}<\alpha^{c_2}(\Delta)$, centered near the point of maximum classicalfrustration, $J_{2}/J_{1} = {1/2}$, for which the intermediate phase exists.This range is widest precisely at the isotropic point, $\Delta = 1$, where$\alpha^{c_1}(1) = 0.44 \pm 0.01$ and $\alpha^{c_2}(1) = 0.59 \pm 0.01$. Thetwo QTP's are characterized by values $\Delta = \Delta^{c}$ at which$\alpha^{c_1}(\Delta^{c})=\alpha^{c_2}(\Delta^{c})$.

Effect of anisotropy on the ground-state magnetic ordering of the spin-half quantum $J_1^{XXZ}$--$J_2^{XXZ}$ model on the square lattice