This is a question some Greeks also struggled with; Aristotle ended up defining children as no more than potential citizens. But this is only because he insisted on a technical definition based on political rights. In practice, Spartan children were indeed citizens, although they were not yet in possession of all their rights or liable for all their obligations; the same is true for underage citizens in modern democracies.

In her recent book Citizenship in Classical Athens (2017), Josine Blok has argued that the most essential and fundamental requirement for citizenship in all Greek communities was birth. In most places, you could only be a citizen if both your parents were themselves citizens. A few poleis also counted the children of one citizen parent as citizens, especially if that parent was male - but this was not the case at Athens or Sparta, at least during the Classical period. In order to be a Spartiate, your mother and father both had to be Spartan citizens. The children of Spartiates and non-citizens were known as nothoi (bastards) or mothakes, and while some of these could gain enormous power and influence (such as Lysander and Gylippos), they were never the socio-political equals of citizens.

At the same time - and again, this was true in all Greek communities, as it is in all modern communities - many citizen rights were only available to adults. Male citizens would acquire political and legal rights, as well as becoming liable to civic and military obligations, when they turned 20. At Athens and Sparta, they would gain additional rights (such as access to magistracies) when they turned 30. At Sparta, a citizen did not acquire the full extent of his rights until he turned 60 and became eligible for election to the gerousia.

None of this meant that they were not citizens until that time. Rather, the future status of the children of citizens was assumed until they reached adulthood, at which time it was often tested. At Athens, citizen children were enlisted in their deme rolls when they turned 18, but anyone could challenge their claim to citizenship and have them stricken from the rolls if they could supply proof. Every now and then the Athenians performed a purge of the citizen body, in which all enrolled citizens had to prove their status or face expulsion. We may assume that something similar happened at times in Sparta.

The difference between democratic Athens and oligarchic Sparta was that at Sparta, mere proof of citizen birth was not enough to secure citizen status. One also had to be able to show that one's property sufficed to meet the contributions to the common messes. In the case of most younger Spartiates, this would have meant in practice that they had to show their parents' estate was good for it. In effect they were not proving the size of their current property, but of their prospective inheritance. However, due to the practice of splitting inheritance between all children (male and female), many Spartiates at some point in their life became unable to meet this requirement, and were consequently ousted from the citizen body.