Answer
If you assume your returns are independent (yes your models might loosen this assumption) then the two models, $Q_1$ and $Q_2$ assign probability distributions to the returns on any given day, $i$: $q_1^i(r^i)$ and $q_2^i(r^i)$.
Presumably you are interested in the model that can more accurately predict the state of the market over subsequent returns, i.e. you are interested in:
$$ \text{Probability of All Observed Returns} = P(r^1, r^2, .., r^n) $$
Under the assumption of indenpendence:
$$ P(r^1, r^2, .., r^n) = P(r^1)P(r^2)..P(r^n)$$
which under the two different models looks like this:
$$ Q_1 = q_1^1(r^1) q_1^2(r^2) .. q_1(r^n) $$
$$ Q_2 = q_2^1(r^1) q_2^2(r^2) .. q_2(r^n) $$
This should be familiar territory if you are used to maximum likelihood expectation. Obviously you would like to choose the model with the highest likelihood. Commonly, to avoid floating point rounding error the maximum of the log is taken since it is a monotonic functions so consider maximising, instead:
$$ log(Q_1) = \sum_i log(q_1^i(r^i)) $$
$$ log(Q_2) = \sum_i log(q_2^i(r^i)) $$
In this case this is also equivalent to the cross entropy between $p$ the true probability distribution which is 1 for the observed state and 0 otherwise, relative to either model $q_1$ or $q_2$. If you do not assume independence of returns then you have a slightly more complicated problem, post more details if otherwise..
Just a thought
If your models are uncorrelated (or have limited correlation) you may be able to improve your accuracy by using an emsemble. Consider the third model:
$$Q_3 = \alpha Q_1 + (1-\alpha) Q_2 \quad \text{for} \quad \alpha \in some[a,b]$$
Now your probability distribution is, $$\alpha q_1^i(r^i) + (1-\alpha)q_2^i(r^i)$$
and ideally you would like to acquire,
$$max_{\alpha} \quad log(Q_3)$$
This will be at least as good as the best model $Q_1$ or $Q_2$ for $\alpha=1$ or $\alpha=0$, but of course you need to cross-validate $\alpha$ otherwise you will just be overfitting this hyper parameter to your observed data.