First, I use two perspectives to answer the question in relation to domestic crimes and the states obligation. Second, I provide an answer to whether the state is obligated for its international wrongs over time and then discuss a few more of the questions that have arisen on this page.
First, Thomas Hobbes argues that to escape the state of nature (chaos and killing) people have agreed to give up their liberty and view of morality in order to have the state's security. Thus, Hobbes would say that a state cannot commit a crime because the state is the sovereign and the states is always morally right and just.
Locke would argue that we give up our liberty to exact retribution for crimes for an impartial justice system. If the state commits a crime we have the right and duty to punish the rulers and their servants because the states only purpose is to uphold impartial justice. (Locke believes the state of nature is that people are social and trustworthy).
These are two different perspectives, conservatism and one of the founders of liberalism.
To understand what Rawls thinks you may use his original position and reflective equilibrium to decipher proper behavior (for a quick view on these two principles http://en.wikipedia.org/wiki/John_Rawls#The_Original_Position). However, in this case who the state is, is not so clear.
[The original vagueness of the question made it a bit difficult for me to answer. First, I was not sure what was meant by a state's crimes? I assumed the inquisitor meant domestic crimes, which I discussed above through Hobbes' and Locke's positions and will leave posted. Second, it was not clear what was meant by moral obligations? After clarifying these points, thank you, I provided the second response.]
Regarding whether the present state is responsible for its past actions:
One element that seems to always constitutes a state as the same over time is its debt. As such, I do not see why that debt cannot take many forms, such as owing others for acts of thievery, mass murder, and cultural destruction.
Who is to blame?
A modern state is both a collection of institutions and rules that extend beyond the power of individuals in society and the individuals in society that legitimate state power and actions. If the state is both, it seems both would be to blame for colonization, genocide, or social welfare. State institutions should take on that debt. However, individuals only share the burden of responsibility in accordance with their personal relationship to power. When individuals who committed past crimes are no longer alive, no individuals share direct responsibility, but because the state and many individuals in the society continue to benefit from the past actions (pilfered lands that help make new investments) they owe in accordance. (my father steals your home and moves it across the river, I take it apart to build myself a home with a different shape and then my father dies. This scenario does not negate the fact that the property was stolen and belongs to you or those you had wished would inherit it).
Who is to repay?
Practically, to condemn everyone in a country is likely not going to solve problems and likely to inflame more. For instance, the way the victors treated Germany after the first world war did not help, and likely fostered, many of the sentiments that propagated the next world war. With that said, individual responsibility is essential for liberalism, civility, and good governance. And punishment is necessary for commitments to these and other endeavors.
Who is morally responsible?
In a sense, morality is the judgement of good and bad, and while good and bad may be contextual to culture and time, those who are interested in liberalism would likely agree that culture and private property are good and their destruction is bad. Therefore, states who do bad things (institutions and the people involved in lending the state legitimacy or who benefited from its bad actions) are obligated to make reparations. If good and bad change, the moral obligation disappears (e.g. nobody owes Louis the 16th, even though many bad french (the poor) stole from the good (the king))