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Just to emphasize it : Depth sorting is not visible on thin curves and is thus useless. If a curve is very complicated, depth sorting may have a significant impact on some hardware, and it is desirable to turn it off.

Also, is there a way to make the depth sorting and antialiasing ControlActive ?

EDIT 3 : Compare the output from the code above, with the output from the same code with the default thickness of curves (code below). The depth sorting is still there, but it is useless since it is not visible from this output :

curve1[t_] := ParametricPlot3D[ {Sin[3 Pi s], Cos[5 Pi s^2], Cos[3 Pi s] Sin[3 Pi s]}, {s, 0.001, t},PlotStyle -> Red] curve2[t_] := ParametricPlot3D[ {1.3 Sin[7 Pi s], 0.5 Cos[2 Pi s], 0.4 Sin[6 Pi s^2]}, {s, 0.001, t},PlotStyle -> Blue] Manipulate[ Show[ {curve1[t], curve2[t]}, PlotRange -> {{-1.5, 1.5}, {-1.5, 1.5}, {-1.5, 1.5}}, Boxed -> True, Axes -> True, AxesOrigin -> {0, 0, 0}, SphericalRegion -> True, Method -> {"RotationControl" -> "Globe"}, ImageSize -> 600 ], {{t, 1, "t"}, 0, 12, 1}]