You've got to remember that in the Greek world, there was only one Sparta. Many states engaged in war, but we remember Athens for its cultural achievements rather than its military successes.
In the Warring States period, virtually every Chinese state was a Sparta. Each state was highly militarized and had a conscription system to enable the equipping and maintaining of hundreds of thousands of men. There were states that went further in this; see Qin as an example, but the point still stands about the environment: how would Greek history have developed if every Greek city-state was actually a Sparta as well, and that each Greek city-state was so dedicated to the elimination of its enemies?
That is the context in which Chinese war developed. After the Warring States period, Chinese society became less militarized and military values began to decline in favor of civilian values, seen in Confucian, Daoist, then Neo-Confucian ideology. This is why the Han army would have trouble against the Romans and that no later Chinese dynasty has been a true military powerhouse; the Ming were barely able to force a truce with the Japanese, a much smaller nation, in the Imjin Waeran and would not have been able to maintain Korea if it weren't for the efforts of the legendary Korean admiral Yi Sun-shin.
Between Qin and Alexander, the main challenge would be the conflict between tradition and genius. I would say that the Qin military system is decisively superior to that of Alexander's, given the context in which it developed. It was capable of complete conscription and had superior military technology, and possibly superior tactics.
Alexander, on the other hand, is sui generis, and if he hadn't died it would have been incredible to see what further victories he would have achieved. His military genius is probably greater than that of any other man of his era or after.
What would have happened is that time would be on Alexander's side. In initial engagements with the Qin army, Alexander would be crushed in every battle he'd choose to keep, because his military system is inferior to that of the Qin's. But as time went on, Alexander would be able to adapt aspects of the Qin military system and understand how to exploit its flaws and weaknesses. As with the Romans, there's nothing stopping Alexander from adopting crossbow technology, even in a debased form, or employing crossbowmen in ranked formation.
On Qin home ground, Alexander would be initially stymied and take heavy defeats at the end of his supply line. If the Qin were able to destroy Alexander there and then, Qin would defeat Alexander as he would lose too many men to adapt Qin weaponry and tactics.
On Greek home ground, however, the Qin would have their own logistical nightmares to deal with; with extremely large forces, it would be very difficult for the Qin to provision their armies and initial battles would not necessary end in their advantage. At the end of the day, the Qin would eventually have to retreat as Alexander would adopt their methods of war and use it to butcher the Qin armies.
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With regards to the crossbow, as someone skilled in archery could tell you, power is not the same thing as range. Range is a function of power and the weight of the projectile.
The most powerful archery weapons tend to be extremely short-ranged, because you need an extremely long power stroke to transfer the full energy of the weapon to the projectile. To compensate, you use a heavier projectile, which can absorb more energy in a shorter stroke, but is ultimately more short-ranged.
Compare the Ming composite bow to the Manchu composite bow, or the Mongol composite bow, for that matter. The Ming composite bow was under the same lines as the Mongol or Turkish composite bow; it was designed for moderate power, a long power stroke, and a light arrow. This allowed it to transfer high amounts of energy to a light projectile, resulting in improved range.
The Manchu composite bow, on the other hand, was a composite longbow designed for horseback use. It had an effective range of only 100 meters, but it could deliver a killing blow to a fully-armored man at short range.
What we know about the Qin armies at the time of Qin Shi Huang suggests that the Qin had a variety of bolts for their crossbows. Some bolts resembled arrows; they were designed to be long and light, meaning that they were good for long ranges and for engaging lightly-armored troops. Other bolts were made completely out of lead, and were intended to defeat armored troops.
The Qin armies then, would be fully equipped to deal with whatever soldiers Alexander could throw at them; armor-piercing bolts would defeat the bronze armor of Alexander's hoplites and even cavalry, while lighter bolts would provide them protection against ranged troops.
I would take issue with claims that the Ge is extremely powerful against Alexander's melee troops, though. IIRC, known Ge tend to be rather light implements, with both a spike for thrusting and an edge for cutting. The traditional use of the Ge, and the reason it died out in Chinese warfare, was to thrust it at the target's head in a feint, then pull back with the cutting edge. The amount of metal and the amount of edge in the weapon was really insufficient for it to be a true halberd, and since it's primarily a cutting weapon, I doubt it would perform well against heavily-armored troops.