Let $H_{\mathbb{Z}}$ be an integral Hodge structure of weight $n$ with Hodge decomposition $H_{\mathbb{C}} = \sum_{p+q=n}H^{p, q}$.
In the definition of a polarized Hodge structure, I have come across two different definitions that differ by a sign (periodic every 4). For example, in Voisin's book it is defined as a $(-1)^n$ symmetric bilinear form $Q$ that is $0$ unless operating on conjugate degrees in the Hodge decomposition, and satisfying
$$i^{p-q}(-1)^{\frac{n(n-1)}{2}}Q(\alpha, \overline{\alpha})>0$$ with nonzero $\alpha\in H^{p, q}$. In various other sources, e.g. http://www.math.columbia.edu/~rf/VHS.pdf, the last condition is simply replaced with $$i^{p-q}Q(\alpha, \overline{\alpha})>0.$$
Can someone explain the discrepancy in the factor $(-1)^{\frac{n(n-1)}{2}}$ between these two definitions?
Edit: Looking closer I realized that in Voisin's definition, the domain is restricted to the primitive cohomology $H^{p,q}_{pr}$. However, my question still remains.