First and foremost, it is not necessary that the Holy Qur’an explains in detail all the concepts. Given that it actually deals with many things, so it does not get involved with the finer aspects and intricacies.
Yet, there is an affirmative reply to this question as well. Many verses of the Majestic Qur’an do prove the existence and also mention about the government of Hazrat al-Hujjat (a.t.f.s.). Eminent scholars have listed such verses in their books. Among them Allama al-Majlisi (r.a.) deserves special mention for having brought numerous such verses in his book “Behar al-Anwar”, some of which are mentioned hereunder:
وَلَقَدْ كَتَبْنَا فِی الزَّبُورِ مِنْ بَعْدِ الذِّكْرِ أَنَّ الْأَرْضَ یَرِثُهَا عِبَادِیَ الصَّالِحُونَ
“And certainly We wrote in the Book after the reminder that (as for) the land, My righteous servants shall inherit it.”(Surah al-Anbiya’, 21:105)
Translator(s): Dr. Sachedina
This book Contains these titles:
BOOK ID؛ point؛ Introduction؛ Chapter 1: The Beginning of the Belief in the Mahdi؛ Chapter 2: The Pseudo-Mahdis؛ Chapter 3: Mahdiism, the Jews, and the Iranians؛ Chapter 4: The Unseen World and the Imam of the Age؛ Chapter 5: Who Was the Imam after Hasan Al-'Askari?؛ Chapter 6: Can a Five Year Old Boy Become an Imam?؛ Chapter 7: Why Did Not the Occultation Become Complete Chapter 8: The Sunni Books؛ ....
Immense oppression that prevails throughout the world, the unjust domination, the hounding of nations around the globe; excessive bloodshed, the disregard of the most noble and the most beloved of human values, this is the treachery that the global supercilious front presents to mankind day after day. Immense immorality, exploitation and oppression naturally strikes at one - who does not have a spring of hope to cling onto - with hopelessness. This is how younger generations in western countries are driven to nihilism. The new generation which has not adapted to nor abides by the traditions, when they see the turbulence of the mankind’s living environment, they lose hope; thus they are driven into nihilism.
As a people awaiting the Imam Mahdi (the Savior), we should establish and build our lives directed upon the same path of our Imam (May hundreds of greetings and praise be upon him and may God hasten his return), today. Of course, we are too trivial to construct the same establishment that the divine friends of God made or will make, but we should work hard towards that path.
The Imam of our time, the Savior (May God hasten his return) is the manifestation of divine justice. We know that the most significant characteristic of Imam Mahdi (God’s greetings and peace be upon him), which has been described in supplications and narrations is justice. “Through him, God will fill the earth with justice and fairness.”
As the successor of the Prophet (PBUH), Imam Ali informs people of an absent Imam from among his descendents who will make world full of justice and remove injustice.
How beautiful it is to combine the event of Ghadir with the words of Imam Ali on his grandson Mahdi.
He is so forgiving
He is a strong shelter
3. Ideal of the good-doers
Whoever comprehends him will find him a glorious light which is an ideal for good-doer
4. Savior of the Mankind
He breaks all chains and terminates the exploitation of man by man
5. Justice Expander
The crowds on earth and heaven are rejoiced by his coming
7. Reward for Steadfast believers
“There will be a long absence for our Imam and beware, those steadfast in their religion whose hearts never darkens by his long absence, they will be on my side in the heaven on the Judgment Day”
8. Praying for Him
“O God! Make his occultation and reappearance the end for all sadness and remove all difference and enmity by him”.
There is a basic question here:
Are we ready to accept the Imam of the Time or our benefits and lust makes us lose the way?!
Are we among those preparing for the green and beautiful future through knowledge and self-education?!
Translated by: Sadroddin Musawi
Promotion of Knowledge and Ethics
Imam Mahdi’s reappearance is the day of endless torment for unbelievers while it is happiness and mercy for virtuous believers. In that era, people’s heart will turn kinder and their morality will be improved. All material thoughts will focus on spirituality.
According to the Holy Prophet Muhammad (PBUH), all people obey and worship Allah and do their best in performing religious practices. In other words, when one perfect and immaculate human being, who is able to teach people to behave wisely, rules over the world, people naturally welcome worshipping of Allah and all moral corruptions will be removed. In addition, if a few disturb and bother others and prevent them from thinking wisely and behaving morally, they will be advised in the first place, else they will be removed. 
Commenting on the same issue, Imam Sadiq (A.S) says: “There are 27 letters of knowledge. The knowledge that has been revealed till date through all the Prophets (PBUT) is the interpretation of only two of those letters. When our Qaem (AJ) will rise, then Allah will reveal the remaining 25 letters and will add to the existing two letters. In this way knowledge with all 27 letters will become complete.”
This Hadith clearly defines the rapid scientific progress of Imam Mahdi’s movement in his era. It also refers to the most powerful and excellent education and training system in that era. Therefore, in that era, the scientific changes and developments will be much more than the previous eras.
Dealing with Offenders
In every society, there are some people who unknowingly commit sins and eventually they are warned through ‘enjoining good and forbidding evil’ which simply convince them stop committing evil deeds. However, there are some others who notice their mistakes only through strict and decisive implementation of divine commandments. Imam Mahdi’s reappearance is the outset of spring for the faithful and winter for wrongdoers.
Quoting the Holy Prophet Muhammad (PBUH), Imam Javad (A.S) states: “He will implement the rules and divine commandments.” 
Translated by: Sadroddin Musawi
1. Al-An’am Surat (124)
2. Bahar al-Anvar, Vol. 52 , P. 336
3. Ibid, Vol. 52, Chapter 27.
Imam Hassan al-Askari (A.S) has said: "There are two characteristics, which surpass all others: having faith in Allah, and benefiting one's brothers (in faith)."
One of the major responsibilities of Imam Hassan al-Askari (A.S), for which he endeavored earnestly, was to introduce the Imam after him to the Shiites and to inform them of the next Imam's occultation. At the time, the Abbasid dynasty, having had heard the words of the previous Imams (A.S) and those of Prophets' (A.S. about the savior that would be born to Imam Hassan al-Askari (A.S), were in close watch of Imam Hassan's (A.S) residency to make sure he would not have a son. Thereupon, Imam Hassan al-Askari (A.S) had to hide his son from the guards sent by the government and in that manner save his son's life. After the martyrdom of Imam Hassan al-Askari (A.S), Imam Mahdi (A.S) went into the periods of his short and long occultation, whereupon most Shiites were deprived from seeing their Imam. Upon this matter, at the time of Imam al-Askari (A.S) it was necessary to introduce Imam Mahdi (A.S) to certain Shiites, and clarify his processes of occultation. Many narrations have been recorded on this matter, but in this excerpt we will refer to only one.
Ahmad ibn Is'haq, one of the reliable companions of Imam Hassan al-Askari (A.S), has narrated: "In one of my visits to the Imam, I had intended to ask about his successor. Yet before I had spoken, the Imam (A.S) said: 'O Ahmad ibn Is'haq! From the creation of Adam, Allah (SWT) has never left the earth without His successor on people, and will never do so until the Day of Judgment…'
I asked: 'O son of the messenger of Allah! So who is the successor and Imam after you?'
Imam Askari (A.S) stood up immediately, and entered the house. He then returned carrying a three year old boy, whose face was filled with a halo of light. He said:
'O Ahmad! If it were not for your eminence in the eyes of Almighty Allah and His chosen people, I would not have shown you my child. His name and nickname are the same as that of Prophet Muhammad's (PBUH); and it is he who will fill the earth with justice after it has been corrupted with oppression…
By Allah (SWT)! He will live through a long occultation from which people will not survive perdition and deviation, except for those who stand firm on believing in his Imamate, and those who are honored with the opportunity to pray for haste in his reappearance."
Ahmad ibn Ishaq narrates: "the minute I asked Imam Askari (A.S): 'Will any sign or miracle appear to make me confident?' the child began speaking:
(Selection taken from," An Analysis of the Life of Imam Hassan Askari (A.S)", by Muhammad Baqir Qurashi)