Inclusion Criteria:

          -  No prior therapy for mantle cell lymphoma (MCL)

          -  MCL in need of systemic therapy, and potentially eligible for ASCT as assessed by the
             treating physician

          -  Documented histological confirmation of MCL by local institutional review

          -  Documented, fludeoxyglucose F-18 (FDG)-avid measurable disease (at least 1 lesion >=
             1.5 cm in diameter) as detected by positron emission tomography (PET)/computed
             tomography (CT) and as defined and includes measurable nodal and extranodal disease
             sites, or splenomegaly measuring more than 13 cm in vertical length

          -  Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1, 2

          -  Absolute neutrophil count (ANC) >= 1000/mm^3 or >= 500/mm^3 if due to lymphomatous
             marrow or spleen involvement (obtained =< 30 days prior to registration)

          -  Platelet count >= 100,000/mm^3 or >= 75,000/mm^3 if due to lymphomatous marrow or
             spleen involvement (obtained =< 30 days prior to registration)

          -  Total bilirubin =< 1.5 x upper limit of normal (ULN) (unless documented Gilbert's
             syndrome, for which total bilirubin =< 3 x upper limit of normal [ULN] is permitted)
             (obtained =< 30 days prior to registration)

          -  Aspartate transaminase (AST) =< 3 x ULN (obtained =< 30 days prior to registration)

          -  Prothrombin time (PT)/international normalized ratio (INR) or partial thromboplastin
             time (PTT) =< 2 x ULN, unless elevated due to a lupus anticoagulant (obtained =< 30
             days prior to registration)

          -  Calculated creatinine clearance must be >= 30 ml/min using the Cockcroft-Gault formula
             (obtained =< 30 days prior to registration)

          -  Negative pregnancy test done within =< 14 days prior to registration for women of
             childbearing potential only

          -  For women childbearing potential (WOCBP, defined as premenopausal women capable of
             becoming pregnant): Must agree to use of highly effective method of birth control
             during study therapy and until 12 months after last dose of study therapy.
             ('Acceptable' methods are not adequate. Highly effective methods are defined by
             Clinical Trials Facilitation and Coordination Group [CTFG] as having a failure rate of
             < 1% per year)

          -  Men must agree to use barrier contraception starting with the first dose of study
             therapy and through 180 days after completion of study therapy

          -  Provide informed written consent

          -  Willing to return to enrolling institution for follow-up (during the Active Monitoring
             Phase of the study)

          -  Hematologic labs must be obtained within =< 14 days of registration

          -  Willing and able to participate in all required evaluations and procedures in this
             study protocol

          -  Ability to understand the purpose and risks of the study and provide signed and dated
             informed consent and authorization to use protected health information

        Exclusion Criteria:

          -  Prior systemic treatment for mantle cell lymphoma. Short course of steroids (=< 7
             days) for symptom management or localized radiation is permissible, as long as
             measurable disease outside of the radiation field exists

          -  Peripheral neuropathy or neuropathic pain of grade 2 or worse as assessed by the
             investigator

          -  Prior exposure to bortezomib or a BTK inhibitor

          -  Prior anthracycline exposure unless cumulative prior exposure is under 150 mg per
             square meter

          -  Requiring anticoagulation with warfarin or equivalent vitamin k antagonist

          -  Uncontrolled AIHA (autoimmune hemolytic anemia) or ITP (idiopathic thrombocytopenia
             purpura)

          -  Active bleeding or history of bleeding diathesis (e.g. hemophilia or von Willebrand
             disease)

          -  History of stroke or intracranial hemorrhage within 6 months prior to enrollment

          -  Requires treatment with a strong cytochrome P450 3A4 (CYP3A4) inhibitor/inducer

          -  Requiring treatment with a proton pump inhibitor. Examples include: dexlansoprazole,
             esomeprazole, lansoprazole, omeprazole, pantoprazole, rabeprazole, or therapeutic
             class equivalents

               -  Note: H2-receptor agonists are not exclusionary

          -  History of allergic reactions attributed to acalabrutinib, cytarabine, bortezomib,
             boron, or any of the other agents administered as part of the therapeutic regimen in
             this study

          -  Active systemic fungal, bacterial, viral, or other infection that is worsening
             (defined as increasing signs/symptoms of infection during screening) or, requires
             intravenous antibiotic therapy

          -  Active or chronic uncontrolled hepatitis B or hepatitis C infection. Patients with
             positive hepatitis B core antibody positive require negative polymerase chain reaction
             (PCR) prior to enrollment. Hepatitis B surface antigen positive or PCR positive
             patients will be excluded. Patients with hepatitis C must have negative hepatitis C
             virus (HCV) ribonucleic acid (RNA) for inclusion

          -  Co-morbid systemic illnesses or other severe concurrent disease (including major
             surgery within 2 weeks) which, in the judgment of the investigator, would make the
             patient inappropriate for entry into this study or interfere significantly with the
             proper assessment of safety and toxicity of the prescribed regimens

          -  Known to be human immunodeficiency virus (HIV) positive since antiretroviral therapy
             has a potential for drug interactions with acalabrutinib

          -  Uncontrolled intercurrent illness including, but not limited to, ongoing or active
             infection, symptomatic congestive heart failure or low cardiac ejection fraction (New
             York Heart Association [NYHA] class 3-4 or ejection fraction [EF] < 45%), unstable
             angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that
             would limit compliance with study requirements

          -  Receiving any other investigational agent which would be considered as a treatment for
             the primary neoplasm

          -  Other active malignancy =< 2 years prior to registration. EXCEPTIONS: Non-melanotic
             skin cancer localized prostate cancer, or carcinoma-in-situ of the breast or cervix.
             NOTE: If there is a history or prior malignancy, patients must not be receiving other
             specific treatment for their cancer

          -  Pregnant and/or breastfeeding

          -  Has difficulty with or is unable to swallow oral medication, or has significant
             gastrointestinal disease that would limit absorption of oral medication

          -  Presence of a gastrointestinal ulcer diagnosed by endoscopy within 3 months before
             screening. unless directly due to MCL Involvement by endoscopic or histologic
             evaluation

          -  Major surgical procedure within 28 days of first dose of study drug. Note: If a
             subject had major surgery, they must have recovered adequately from any toxicity
             and/or complications from the intervention before the first dose of study drug

          -  Concurrent participation in another therapeutic clinical trial