Why quantum spin liquid has negative Curie-Weiss temperature? In a table in wikipedia, the Curie-Weiss temperatures of quantum spin liquids are listed. Among them, the $\Theta_{cw}(K)$ are less than zero. Why are they negative? Since temperature is defined as 
$$T = \frac{dq_{rev}}{dS}$$
I would expect the additional heat let the spin liquid tend to disorder. 
 A: Naively, both temperatures (Curie and Curie-Weiss temperature) are equal and they're the constant temperature $T_c$ entering the Curie-Weiss Law:
$$ \chi = \frac{C}{T-T_c}. $$
However, the behavior is often more complicated and the formula above doesn't describe the susceptibility $\chi$ well for all temperatures. When it's so, the Curie temperature $T_c$ is the temperature at which the susceptibility actually blows up, so $\chi=C/(T-T_c)$ holds for $T\sim T_c$ while the Curie-Weiss temperature is the temperature for which the law $\chi=C/(T-T_0)$ holds for $T\gg T_0$, i.e. one reconstructed from the "shape of the hyperbola far away".
The temperatures are close $T_0\sim T_c$ for materials for which the transition is first-order; the temperatures are very different if the transition is second-order. Since it's the Curie-Weiss temperature that may be negative and it is extracted from the behavior of a curve far enough from the actual transition, and the behavior may be rather complex, your simple positivity argument (more heat means higher entropy) isn't really able to determine the Curie-Weiss temperature, not even its sign.
At any rate, the Curie-Weiss temperature is just a constant with the units of temperature that happens to enter some more complicated law. We are actually not adjusting any physical system to this temperature – it is not possible if it is negative – so the Wikipedia page about the negative temperatures isn't really relevant. We're not studying any system at a negative temperature; just a system at a positive temperature, much higher than $T_c$ or $T_0$, where the behavior reproduces some curves that may be parameterized by the Curie-Weiss temperature. Something special would occur at the Curie-Weiss temperature if the hyperbolic shape of $\chi$ could be extrapolated. But it cannot so all the special things actually occur at positive temperatures.
I am not saying that "negative temperatures" are indefensible as a generalized concept at all situations; I am just saying that this generalization isn't needed when discussing the Curie-Weiss temperature, not even if the latter is negative.
