chi-squared-distributionnotationquantiles
Please take a look first at this link: http://en.wikipedia.org/wiki/Ljung%E2%80%93Box_test#Formal_definition
It is written, $\chi_{1-\alpha,h}^2$ is the $\alpha$-quantile of the chi-squared distribution with $h$ degrees of freedom.
Does $\chi_{1-\alpha,h}^2$ mean:
Following the logic of the case with 1 df, in the case of 2 dfs we can also be expressed the threshold as the 97.5 percentile of a N(0, ~1.25) Normal distribution, which is about the square-root of 6.0.
In the 1 df case it doesn't just match at the 97.5 percentile -- all of the percentiles correspond in a similar way.
While there's definitely a connection in the 1 df case, this is not really relevant for other degrees of freedom. How did you arrive at 1.25 (aside from matching its 97.5 quantile to $\sqrt{6}\,$)?
You could as easily say "well, see it's connected to a logistic distribution" and then compute the corresponding scale parameter for a logistic by matching two quantiles in a similar fashion. That doesn't imply they're related in any meaningful sense.
For the chi-squared(1), the two corresponding curves lie one atop the other, rather than just crossing somewhere.
The sum of two chi-squared(1) variates is exponential with mean 2. You can match any individual quantile to some normal distribution but you can't choose a single standard deviation that will make all the quantiles correspond at the same time like you can with 1.df
e.g. The 75th percentile of a chi-squared(2) is about $1.67^2$; to get the 87.5 percentile of a zero-mean normal to be 1.67 you need $\sigma$ to be about $1.448$. Would you want to use a different $\sigma$ for each quantile? I don't see much understanding to be gleaned from that activity.
Instead of asking what the difference is between the two, it's clearer to ask what the relationship is between 𝜒2 distribution and the Probability-To-Exceed (PTE).
The PTE is the probability of obtaining a higher 𝜒2 than what you actually achieved. 𝜒2 is a measure of how far off your values are from expectation, and a higher value means larger disagreement. A very low PTE means it is very unlikely to get a higher 𝜒2 than what you already have, meaning your values are farther off from expectation than random chance would allow. In the opposite extreme, a very high PTE means it is very likely to get a higher 𝜒2; this is also bad because it usually means you have overestimated the errors on your measurement.
To calculate the PTE, integrate the 𝜒2 distribution up to your value of 𝜒2, and subtract that value from 1. Usually this is done via a look-up table or solved numerically with a computer program, since there is not a closed-form solution.
The quoted text then goes further, wanted to relate this PTE into a "sigma" of a gaussian distribution since that is a more commonly understood metric.
Best Answer
I'm not surprised this is confusing.
Usually the subscript "$1-\alpha$" indicates the chi-square value is at the $1-\alpha$ quantile of the distribution.
So that would suggest it has an area $1-\alpha$ to its left and $\alpha$ to its right.
That is to say, your option 1.
But on the other hand "$\alpha$-quantile" would more usually suggest the lower tail, your option 2.
So which is it:
Well, the statistic should reject when correlations are large, and the formula will produce large values of the test statistic when correlations are large.
So when the population correlations are zero, the upper tail of the null distribution, of area $\alpha$ forms the rejection region, so you need $\alpha$ in the upper tail.
Which is your option 1.