In Quran we have a lot of verses talking about the hell and the heaven and their characteristics.
What I want to know is that whether the hell and heaven mentioned in Quran currently exist?
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There are Quranic verses implying that heaven and hell currently exist. But note that Quran uses the word 'janna' (Eng: garden or paradise) to describe the blissful place that hosts believers in afterlife.
This, in Christian scriptures, is predominantly referred to as 'heaven', although the word Paradise is fairly common, too.
Also noteworthhy is that the word janna very often is followed by descriptive nouns such as mawa (Eng. repose), adn (Eng. eden), firdaws (Eng. paradise), etc to specify different levels of Paradise. Some examples can be seen in 32:19, 9:72 and 18:107.
Also, in 53:12-15 which mentions the story of Prophet's Night Ascension or Miraj, Quran says that Prophet saw Angel Gabriel "at the Lotus Tree of the Utmost Boundary (or Extremity) near which is the Garden of Repose."
This specially interesting Quranic verse talks about an otherworldly 'tree' existing in near an otherworldly 'garden', concepts that are common to Quranic description of Paradise resembling gardens of trees. It is clear from the verse that Garden of Repose has been already in existence when Prophet encountered it in his Miraj. And if this garden is a part of the general place that hosts the good believers (for which there's no indication to the contrary), then it also follows that Paradise is already created and in place.
There's also a hadith from Prophet Muhammad indicating that particular conditions of believers in heaven or hell are immediately shaped by their deeds in this world. That implies that heaven and hell already exist independent of this world but the conditions of the specific residence of each person in them are constantly influenced by their deeds in this world. (I will reference the source once I could locate it.)
There is a plenty of intricate suitabilities to this fascinating topic which have been a subject of debate and wonder for muslim theologians and philosophers. Uniquely intriguing and profound on this subject are the theories of Mulla Sadra the genius of Islamic Metaphysics. For those interested I recommend studying this paper, The Soul as Barzakh: Substantial Motion and Mulla Sadra's Theory of Human Becoming.
Heaven and hell have been made by Allah ,Glorified and Exalted be He, to reward and punish humans.
As to current references in Quran regarding heaven and hell, they obviously have to be referred in present tense so that they can affect/change us for good in the present lives. These are the things of unseen ( Ghaib ) which no one knows except Allah talah and hence no scholarly article can confirm/reject it.
The answer to your question is we cannot know for sure. All ayas about jannah are allegorical (mutashabe) only our lord knows their precise meaning.
هُوَ الَّذِي أَنزَلَ عَلَيْكَ الْكِتَابَ مِنْهُ آيَاتٌ مُّحْكَمَاتٌ هُنَّ أُمُّ الْكِتَابِ وَأُخَرُ مُتَشَابِهَاتٌ ۖ فَأَمَّا الَّذِينَ فِي قُلُوبِهِمْ زَيْغٌ فَيَتَّبِعُونَ مَا تَشَابَهَ مِنْهُ ابْتِغَاءَ الْفِتْنَةِ وَابْتِغَاءَ تَأْوِيلِهِ ۗ وَمَا يَعْلَمُ تَأْوِيلَهُ إِلَّا اللَّهُ ۗ وَالرَّاسِخُونَ فِي الْعِلْمِ يَقُولُونَ آمَنَّا بِهِ كُلٌّ مِّنْ عِندِ رَبِّنَا ۗ وَمَا يَذَّكَّرُ إِلَّا أُولُو الْأَلْبَابِ
It is He who has sent down to you, [O Muhammad], the Book; in it are verses [that are] precise - they are the foundation of the Book - and others unspecific. As for those in whose hearts is deviation [from truth], they will follow that of it which is unspecific, seeking discord and seeking an interpretation [suitable to them]. And no one knows its [true] interpretation except Allah . But those firm in knowledge say, "We believe in it. All [of it] is from our Lord." And no one will be reminded except those of understanding.